Are there countries where all homeowners associations are prohibited
from exercising control over non-common property?
Introductory points
For the most part, the issue is not the country in which the homeowners associations are located. It is the terms of the governing documents of those "common interest communities" in a particular association that matter. This answer explores why these governing documents may have different terms in different countries after exploring their U.S. history in some depth.
This answer largely lays out a stylistic "lies to children" historical narrative that hits the big ideas in a way that accurately captures what was going on, and how the U.S. got to where it is today, but in an oversimplified manner that glosses over all but the most glaring regional differences and differences between states expressed in state specific legislation.
The issue
Analytically, you can (and HOA governing documents often do) break things down into several categories:
Property that is owned by the HOA outright (like walkways and grassy common areas and club houses and driveways).
Property that forms the common physical infrastructure of multiple units (often these are classified as "limited common property" that may not serve all units in the community. For example, elevators that serve one wing of a building, structural walls, roofs over a building that has some units in it, shared water and sewer lines (even if individually metered), etc.
Rules and regulations that are necessary to the operation of common property and limited common property (e.g., not engaging in activities that are fire or safety hazards, allowing access for maintenance, not blocking or damaging common driveways or common or limited common property). Some level of architectural review of projects that could impact common or limited common property is usually necessary in connection with this (e.g. approval of proposals that could disturb a structural wall).
Rules and regulations that are not necessary for the operation of common property and limited common property in a physical sense, but are considered desirable to make the HOA a nicer place to live (e.g. rules banning noisy activities after certain hour, rules banning Airbnb rentals, rules banning or requiring approval of potentially ugly exterior finishes, rules requiring members to keep landscaping to certain standards, rules requiring broken windows and graffiti and property damage that doesn't affect other units to be fixed promptly because it is an eyesore). These are sometimes called "restrictive covenants".
As I understand the person asking the question, the issue is, why are there not more HOAs that do (1), (2) and (3), but not (4).
The U.S. experience
In the U.S. context, this is driven mostly by economics and custom, by local politics, and by U.S. Constitutional law, rather than by legal prohibitions on establishing HOAs that do (1), (2) and (3) but not (4). And, I understand very well how aggravating (4) can be in the hands of a bad HOA board or a contentious set of HOA members. And, let's set aside for the moment the non-obvious question of which regulations are on which side of the boundary between category (3) and category (4).
The issues from local politics (and related issues of U.S. Constitutional law) related to the fact that often, at least when they are initially established, HOAs are established in the U.S. as municipal government substitutes in land that is not part of an incorporated municipality.
The U.S. local government context: county government
In the U.S., almost all land has local county government. This government is typically governed by 3-5 county commissioners, elected at large in the county or by single member districts, in a government which has lots of responsibilities that are mandated for it by state government but administered locally, like providing law enforcement and a jail serving territory outside of municipal government jurisdiction through an elected sheriff, maintaining real estate records and collecting property taxes for all local governments in the county, operating a courthouse and helping to finance the local prosecuting attorney's office, administering state welfare programs, child protective services, building and maintaining state and local roads, handling vehicle registrations and inspections, carrying out some public health and environmental regulations, administering building codes in areas outside municipalities, establishing and administering basic zoning codes outside municipalities, and administering federal soil and water conservation grants for agricultural land. Sometimes they perform other services associated with municipal government, and sometimes they don't. Counties usually can't impose income taxes or miscellaneous taxes (like head taxes), and are less likely to impose development fees or sales taxes (or impose smaller ones) than municipal governments.
The U.S. local government context: municipal government
Other government services and functions are often only provided by municipal governments which don't necessarily match county lines and typically only include developed real estate. These include libraries, parks, fire protection, law enforcement limited to and controlled by the municipality, noise regulations, homeless shelters, public art, water and sewer systems, granting cable TV franchise to a single company, negotiating which utility companies can serve residents, and transit systems. Municipal governments also generally have broad authority to impose a wide variety of taxes and to impose a wide variety of generally applicable laws (e.g. minimum wages, landlord-tenant regulations, etc.) with no fixed boundaries. Municipal governments also tend to have more rigorous zoning and building code regulations of land use.
Exceptions for New England and Hawaii
An important exception to this in the U.S. is New England, where almost all land is not only part of a county government (which tends to be particular anemic) and a New England town government which is similar to a municipal government (when it is not in a true incorporated city). Another is Hawaii, where the state government performs functions that would be municipal government functions in most states.
Developers prefer HOAs and special districts to municipalities
This brings us to the economics and custom part of the equation.
In the U.S., real estate developers are the people who form almost all municipalities (since at least the homestead era in the mid-1800s), and almost all HOAs and special district governments (since the 1960s). Developers have overwhelmingly voted with their feet to prefer HOAs and special district governments over municipal governments since at least the 1960s or so. This is illustrated, for example, in a Politics.SE question and answer about local governments in the U.S. state of Illinois.
This is time period that corresponds to growing U.S. Constitutional law restrictions on what governments can do, greater use of restricting zoning and building codes, widespread suburban development driven by the build out of the U.S. interstate highway system, and authorizing legislation for HOAs and special districts in many states.
Why do developers prefer HOAs to municipal government?
Many real estate developers prefer to build developments outside of municipal boundaries where they face only county level regulations rather than both county and municipal regulation, substituting HOAs to provide services and regulations where they can, and special districts which much more limited taxing and regulatory authority to provide services where they can't (e.g. fire protection). They often find this arrangement to give them more control over how their real estate developments are set up, and more insulation from municipal taxes (in addition to county level taxes payed by everyone).
Another factor is that U.S. Constitutional law imposes lots of limitations on what governments can do, but far fewer limitations on what private organizations can do. So, for example, an HOA can legally impose regulations that limit speech in a content based way, while municipal governments cannot do that. In the U.S. it takes much more expertise to run a municipal government lawfully than it does to run an HOA lawfully.
Real estate developers are concerned about the short term from acquiring land and "entitling it" (i.e. getting legal authorization to build their projects), until all the units in a development are sold, which typically spans three to ten years. During the early part of this time period they want maximum autonomy and minimum taxation. During the later part of this time period they want maximum control until their development is sold out and want to maximize the sale price of what they are selling.
Being outside of a municipality is part of the marketing proposal that is used to favor sales in their developments over nearby urban housing, by claiming that buyers will have fewer taxes and fewer government imposed rules by "untrustworthy" "low class" city dwellers who are too liberal.
The restrictive covenants and services of the HOA are used to market a vision in which buyers aren't giving up services by living outside a municipality and are held to enforceable legal standards that prevent activities and poor homeownership conduct associated with "bad neighborhoods" in cities from affecting their more utopian development where everyone is upstanding and behaves well. Developer control of the HOA until most properties in the development are sold allows the developer to make the promise of the benefits of these restrictive covenants seems credible.
Aesthetic and speech restricting restrictions on land use are particularly attractive to real estate developers, because here HOAs provide elements of their utopian residential development that cities are not legally permitted to provide because they are governmental entities. This "comparative advantage" of HOAs makes covenant particularly common in residential HOAs outside of incorporated municipalities.
But, since the developer is around only for the short to medium term, the developer isn't very interested in the long term viability of the HOAs put in place once they are turned over to resident elected HOA boards that aren't made up of real estate development professionals.
Long term problems with HOAs
The dirty little secret of HOAs (and small local governments) is that the smaller the population from which a governing board is elected is, the less competent the board managing it will be, on average, and the more prone the board will be towards acting based upon personalities and prejudices instead of larger good policy considerations.
But, as long as the HOA is not an immediate abject failure once turned over to a board of fresh enthusiastic new residents of a new development, the developer doesn't really care how it works out decades later when the developer is long gone.
Decades later, HOAs with restrictive covenants often are (even usually are) poorly governed, relative to HOAs with less restrictive covenants. But, at the time of formation, the more restrictive the covenants, the easier it is for the developer to market homes in the HOA as part of a utopian community. And, when enough developers act on this set of incentives enough times, this approach to HOA governing documents and restrictive covenants becomes customary and is repeated thoughtlessly over and over again.
In response to this there is an emerging, but still weak, trend to restrict HOA authority and more heavily regulate HOAs in particular narrow areas where legislators have been made aware of particularly concerning abuses and problems.
For example, many state laws require HOAs to allow at least some political signage, and there are U.S. Federal Communications Commission (FCC) regulations that prevent HOAs from banning small satellite TV dishes for HOA units (often adopted as covenants back when satellite TV dishes were 2-3 meters in diameter, and never changed).
There are also hidden problems with seemingly the seemingly neutral approach of democratic majority rule of HOA boards as opposed to ownership by a single entity or landlord that has contractual obligations to unit occupants.
For example, suppose that you are in a five story condo building. The median voter for an HOA board is on the third floor, and the hardship faced by residents on the first few floors if the elevator doesn't work due to repairs is modest for them, but extreme for residents on the higher floors. So, the HOA board will tend to underinvest in paying elevator maintenance people to do their work on an expedited basis. In contrast, a single owner with potential breach of contract liability to residents on the top floor may invest a more appropriate amount on making elevator repairs take place as quickly as possible.
More generally, it is much more difficult to compel a group of board members or HOA members to appropriate funds for the needs of a minority of HOA members than it is to do so when the entire development is controlled by a single individual who has contractual obligations to all of the members.
Which aspects of HOA development patterns more common in the U.S.?
Parts of this U.S. history are specific to the U.S. that do influence how governing documents tend to be drafted in the U.S. v. many other countries.
many countries require almost all land to be part of a full fledged municipal government and not a local government of only limited powers like a U.S. county, and don't permit developers to freely create special districts to undermine this system;
some countries don't make as stark distinctions between the significantly regulated authority of governmental entities in the U.S. and the lightly regulated authority of private HOAs in the U.S. (other countries tend to restrict governmental power less than the U.S. and restrict HOA power more than in the U.S.);
some countries have municipal governments that regulate matters that are regulated by HOAs where they are regulated at all in the U.S. (like what color you can point your house), sometimes through historic land use regulation since many countries have urbanized areas that are much older than the U.S.; these countries tend to have more infill development and less greenfield development than the U.S.;
many countries didn't experience the wave of greenfield development driving by highway construction and vast amounts of undeveloped land that the U.S. fueled the HOAs/special district strategy in the U.S.;
Japan handles land use regulation mostly at the regional level rather than at the local government level, and affords less autonomy to local governments than the U.S. does, which impacts these incentives;
in many countries HOAs that exist are a product of conversions of multi-unit residential buildings built and owned by individual/corporate landlords or by government agencies into separate ownership by individual unit owners, rather than new real estate developments built from scratch to be part of HOAs.
What parts of the HOA development process are universal?
But, this being said, it is still the case that the choices made by someone establishing an HOA in the governing documents is typically more important than the laws of a country, in determining how much category (3) v. category (4) regulation is done, and how often HOAs are created when they aren't strictly necessary for category (2) issues (e.g. many U.S. HOAs involved neighborhoods full of single family homes, which are more rare in most countries).
Real estate developers and people converting single owner multifamily residential properties to individual unit ownership still want to portray a utopian vision of what it will be like to live in those units that credibly rules out factors associated with undesirable neighborhoods (e.g. by ruling out building standards and conduct associated with poor people). And, both real estate developers and property converters also still only care about the short to medium term until their units are sold, not the long term success of the HOA.
Select non-U.S. experiences
In places that were already highly developed and urbanized before HOAs came into wide use, such as Western Europe and Japan, this tends to produce fewer HOAs that aren't a practical necessity for managing multi-unit housing, and less of a focus on restrictive covenants.
The answer from Erik Hart regarding the German version of HOAs exemplifies this path of development of this kind of institution. On part of the narrative the Erik Hart's answer doesn't get into as much (because it isn't directly responsive to the question) is that laws of general applicability in Germany regulate daily life there in a much more invasive way in many respects than U.S. law does in HOAs with lots of covenants.
For example, in Germany, there are days of the week and times of time, where strict noise restrictions are in place and you can't even conduct do it yourself construction or craft type work even in your own home. Business hours are much more heavily regulated and the degree to which government licensing and red tape is required is much greater for many occupations (which would have no comparable U.S. licensing like working in lower level restaurant jobs) and for many kinds activities (e.g., driving, having employees, disposing of trash, leasing real estate, hunting, fishing). Stricter generally applicable regulations reduce the need for HOA specific covenants.
One of the comments to the question, from Italian Philosophers 4 Monica notes that:
stratas in British Columbia are, since 2022, mostly prohibited from
enforcing age covenants on residents. So this is a case where an
agreed upon contract/agreement between private parties gets an
override from the government.
But, there are some places, e.g., in India, where new developments with HOAs much more powerful than those in the U.S. that are basically true privately owned cities exist.
There are also lots of places, especially in developing countries like Mexico and Nigeria and South Korea, where existing municipal governments are so underfunded and impotent, that stronger governance and regulation through an HOA (frequently in gated communities) can be quite attractive, even if these kinds of developments are much less common there.
For example, Mexico (and most of Latin America) has very little land use regulation, and more generally, weak and underfunded local government.
Several of the comments to the question discuss at some length the relative absence of HOAs with single family homes in most non-U.S. countries for reason that the answer above should help explain, although it isn't entirely the case with gated communities in much of the developed world also having this character.
Post-script regarding formalities of organization
Some of the comments to the question explore at some length whether HOAs are contractual, are a form of entity, or are something else. The legal mechanics vary.
But the formalities are mostly a means to an end, and with the exception of the limitations on enforcing restrictive covenants in British Columbia, Canada, are not very determinative of the form of restrictive covenants. The question, rightly, focuses on the practical rights of the HOA to regulate how members use their property and the exact legal form of the common interest community doesn't matter all that much with respect to that point.
Post-script re residential co-ops in the U.S.
The earliest forms of democratically owned multi-family housing in the U.S., mostly in the Northeast U.S. in major cities on the Eastern seaboard, were "co-ops". They had and continue to have a different formal organizational framework than condominiums and more modern HOAs that benefitted from state authorizing legislation. Coops have state legislation now but didn't when they were were devised and became common. In particular, in the days of early U.S. residential co-ops, ownership of single buildings with multiple residential units couldn't be legally subdivided, so the only alternative was to rent from a landlord woh owned the whole building.
One of the defining aspect of Northeastern U.S. residential co-ops is that the exercise tight control over who could and couldn't become unit owners, because all owners were co-guarantors of a common mortgage for the building. This made the risks involved with having even slightly "eccentric" neighbors even they could currently afford to buy into the co-op much riskier.
This is mostly a regional historical footnote in the U.S. today, and some co-ops have converted to condominiums with typical modern HOAs. But this tradition also influenced the idea that a condominium or HOA of single family units could have strict covenants, in the legal tradition before HOAs became common.