Unlike the other answers, which aren't false but don't capture the entire story, I would explain what we observe mostly in terms of politics rather than policy.
Disclosure At The State Level
In the U.S. system of federalism, the laws related to the formation and organization of business entities and trusts is governed primarily by state law, rather than federal law.
Also, under the full faith and credit clause of the United States constitution, an entity formed in one state must be recognized in other U.S. states as well. For example, even if your business is located entirely in Ohio, you can incorporate in Nevada or Delaware and be subject to that state's laws regarding the internal affairs and ownership rules for an entity rather than the laws of Ohio.
This has given rise to what is known as the "race to the bottom" in which states fall over themselves to have organizational statutes that are friendly to the whims of would be incorporators. And, one thing that would be incorporators often like is a lack of public disclosure of who owns the business.
As part of the race to the bottom, disclosure of ownership requirement have almost completely vanished (or was never imposed in the first place) for both business entities and trusts.
Notably, in areas where "race to the bottom" approaches aren't available, such as in regulated industries like law firms, licensed real estate companies, companies with liquor licenses, and companies that participate in state regulated marijuana sales, state distaste for disclosure of beneficial ownership disappears and states routinely require full disclosure of an entity's owners (and sometimes their debt investors as well) in publicly available filings.
Disclosure At The Federal Level
Notably, this trend is not seen at the federal level, which regulates all entities for the limited purpose of tax law, and regulates disclosure requirements pertinent to publicly held companies and disclosure requirements for entities that file lawsuits in federal courts.
Disclosures Of Ownership For Federal Tax Purposes
For federal income tax purposes, (1) all owners of entities organized as single member LLCs, (2) all entities taxed as partnerships under Subchapter K of Title 26 (the Internal Revenue Code) such as general partnerships, limited partnerships, and if they do not elect to be taxed as corporations: limited liability partnerships, limited liability limited partnerships, limited liability companies, and limited partnership associations, (3) all entities taxed as corporations which make elections under Subchapter S of the Internal Revenue Code, (4) all grantor trusts, (5) all simple trusts,
(6) all complex trusts that make distributions to beneficiaries, (7) all private foundations, (8) all REITs, (9) all mutual funds, and (10) a C-corporations that pay dividends in a given year, must disclose their owners/beneficiaries to the IRS in connection with their annual information tax return requirements.
These disclosures are not available beyond the IRS and other people who can subpoena a tax return or who have a right to review a tax return as a co-owner of a closely held business, but they also mean that beneficial ownership isn't unknown to the government in these cases.
Pretty much the only entities that do not have to report their owners to the IRS are (1) complex trusts that distribute nothing to beneficiaries in the current year, and (2) C-corporations that do not pay dividends in a given year.
It is really even more pervasive than that, because there are significant tax disincentives to create C-corporations except in cases of foreign ownership or publicly held companies, where C-corporations are the only permitted tax regime. For example, there are strong tax reasons for C-corporations not to own real estate, where the tax treatment of pass through entities who disclose their ownership to the IRS is much more favorable.
From an economic perspective, the lion's share (perhaps 90%+) of C corporations measured by revenues, income, assets, employees, or corporate taxes paid are publicly held companies that have to disclose their ownership structure in any case to a great extent under the securities laws (although disclosing the ownership structure of C-corporations that don't pay dividends isn't that important from a tax collection perspective because those corporations pay income tax at the entity level rather than the ownership level).
In 2014, in the U.S., there were 303,000 complex trusts that didn't have to report their beneficiaries to the IRS because they made no distributions in the current year out of 3,171,000 trusts and estate in all (rounding to the nearest 1000) and some of those had to report beneficiaries in prior years when distributions were made by those trusts to beneficiaries.
There were 3,613,000 entities taxed under subchapter K who had to report their partners to the IRS.
There were 4,257,909 S corporations who had to report their shareholders to the IRS.
There were 24,631,831 non-farm sole proprietorships, a category that includes LLCs with a single owner that had to report their ownership to the IRS.
There were about 1.6 million C corporations in the U.S., a category that does not have to report beneficial ownership to the IRS unless dividends are paid in the current year. About 1% of these are publicly held companies that make of the predominant economic share of the total. Only 819,000 of them have any taxable income (most of those without taxable income pay their taxable income to a shareholder-employee as a bonus each year which is reported to the IRS as wage and salary income), and all but about 200,000 either have no income tax due as a consequence of tax credits (with the balance paid as compensation to shareholder-employees in most cases), or have income that is due only in lower tax brackets for small corporations that are lower than the tax brackets for publicly held companies and individual income earners.
For example, 82.2% of corporations that owed any corporate income taxes owed just 0.6% of corporate income taxes owed, and 93.5% of corporations that owed any corporate income taxes owed less than 2% of all corporate income taxes. Meanwhile, fewer than 40,000 corporations (at least a third of which, if not more, are publicly held) owe 98.8% of all corporate income taxes, despite being just 0.6% of all corporations.
There are about 37.3 million businesses and trusts in the United States. All but about 303,000 trusts and about 1.6 million C corporations don't have to report their beneficial ownership to the IRS in years when they make no distributions to their beneficial owners, the lion's share of the C corporations economically are publicly held companies that have to make disclosures anyway, and the lion's share of the privately held C corporations aren't making distributions to shareholders because they are making distributions to employee-shareholders who are disclosed to the IRS in their capacity as employees.
Only about 2% of business entities or trusts in the U.S. have economically significant beneficial owners who are not known to the IRS as persons economically affiliated with the business either as owners or as employees. These are mostly concentrated in about 303,000 complex trusts that make up about 10% of all trusts in existence, and about 200,000 medium sized privately held C-corporations that don't distribute dividends to shareholders on a regular basis and collectively aren't very economically important in the context of all outstanding business activity in the U.S.
Privately held C-corporations are required to have less than 300 shareholders, but the median number of shareholders in a privately held C-corporation is less than 4, and something on the order of 95-98% have ten or fewer shareholders.
There is a huge hole in the U.S. economy of companies with more than a dozen but less than several hundred owners, with the primary exception of limited partnerships that own either oil and gas interests or other kinds of real estate investments, which are usually organized as Subchapter K limited liability entities that must disclose their owners to the IRS each year.
Federal ownership disclosures requirements for publicly held companies.
Furthermore, publicly held corporations must disclose all owners of at least 5% of the outstanding shares in the public record and can be compelled to make ownership records available to someone seeking to bring a proxy fight which in entities with thousands of shareholders each and the ability for anyone to buy shares without approval from anyone else, amounts to de facto public disclosures of beneficial ownership (at least of the people or entities who directly own shares). Again, these requirements are imposed by the Securities and Exchange Commission, which is a federal agency.
Disclosure of the ownership of federal litigants
As a third example, any entity that enters an appearance in a federal court case must disclose its beneficial ownership, but few state court systems have an ownership disclosure requirement.
The bottom line is that at the federal level, where a "race to the bottom" isn't possible, there are rather pervasive requirements for entities to disclose beneficial ownership to someone or other, usually for either tax purposes or securities market regulation purposes.
But, at the state level, where a "race to the bottom" has been the rule almost as long as it has been possible to create entities without express legislative permission (something that happened in most states sometime in the 19th century), there have not been ownership disclosure requirements, even though disclosure is required at the state level in circumstances where the "race to the bottom" is not applicable.
Given the fact that the federal and state governments are regulating the very same entities in parallel, have the same constituents, and often are governed in the federal case by former state government office holders whose campaigns are managed by the same political party organizations, it is much more plausible that disclosure requirements are a product of federalism and the "race to the bottom" mechanism of business entity and trust law which is made at the state level, rather than mostly as a consequence of bona fide policy considerations as the other answers to this question suggest.