I think the law doesn't regulate (or only indirectly regulates) the insurance providers.
The main point is that it regulates the service providers (i.e. the health care providers, e.g. doctors).
I think the law is that you (a health care provider) are legally not allowed to charge privately (e.g. to charge a patient or a private insurance company) for any type of service (e.g. surgery) that would be paid for by the government's universal health insurance (which includes any or every kind of mainstream health care).
So it's not especially that there's no private health insurance, it's that there's no private health care (people choose their own doctors, but the doctors and hospitals are paid by the government).
And the rationale for that restriction (i.e. no private health care) is that they (the government i.e. the people) don't want a "two-tier" health system, where rich people (and/or people with private insurance) get better health care, and other people get worse health care.
I think a Canadian Health Minister (i.e. a politician) once said that in the States anyone can get however much health care each person can afford, whereas in Canada everyone can get however much they need.
I think I heard that leads to cases where e.g. a professional athlete goes to the States to get a soft-tissue injury looked at quickly (and at private cost), but generally I don't think that (i.e. cross-border shopping) is a big factor.
Private insurance in Canada can be obtained for things whose whose cost is not covered by the universal (provincial) plans -- including e.g. prescriptions and prostheses (the universal plan pays for medical services), dental care, eye glasses, maybe physical rehabilitation, pay your lost wages, travel insurance, cosmetic surgery, that kind of thing, maybe a private room in a hospital -- that insurance is often a (taxable) "benefit" paid by your employer. But there's no insurance to pay for surgery or to visit a family doctor, because those costs and services are covered by the government plans.
Or these "supplementary" costs listed above (e.g. prescriptions) which aren't covered universally may be covered by "welfare" payments, e.g. based on means-tests for people with no or low incomes, or people who have catastrophically high prescription drug costs.
A further result may be (I don't know) that the amount a service provider (e.g. a doctor) can charge is lower, too, that it might be otherwise -- since the government has a monopoly on paying them.
I think I read that the costs of prescription drugs, too, is lower in the Canada than in the States, even though people pay for those out of their own pockets. I'm not sure why that's so, I think it's that the government can (and must) to permit whatever drugs are sold and that it negotiates a price with the drug companies, and that a new drug which is too expensive isn't permitted -- but that's a whole other topic.
Wikipedia (however I'm not sure that this tells the whole story):
Six of Canada's ten provinces used to ban private insurance for publicly insured services to inhibit queue jumping and so preserve fairness in the health care system. In 2005, the Supreme Court of Canada ruled that in Quebec, such bans are unconstitutional if the waiting period for care is excessively long. However, this ruling only applies within the Province of Quebec. A second court challenge is currently underway to determine whether the prohibition of private parallel health care violates the patients' right to life, liberty, and security under Section 7 of the Canadian Charter of Rights and Freedoms.
The criticisms section is long -- but perhaps that's failing to talk about the good news; or the good news is easily summarised:
Health cards are issued by provincial health ministries to individuals who enroll for the program in the province and everyone receives the same level of care. There is no need for a variety of plans because virtually all essential basic care is covered, including maternity
I don't think the service is much worse than elsewhere, and think that it's more-or-less comparable to other developed countries -- but is perhaps unusual (compared to e.g. the UK or France) in not even permitting private health care.
Based on the article The illegality of private health care in Canada written in 2001, which was added to the OP, perhaps I should revise my answer:
- What I said is true of some but not all provinces
- Details vary, e.g. in some provinces (e.g. Ontario) private health care is permitted but a physician isn't allowed to charge more than they'd be paid in the public system
- In other provinces a physician may charge more but if they do then they're not paid by the government at all (i.e. the user pays the whole cost not just the difference), and private insurance isn't permitted.
It says ...
We conclude by noting that in Canada, the absence of a private system is not due to the illegality of private health care per se. Private insurance for the kinds of medically necessary hospital and physician services that the public service is meant to cover is illegal in only 6 provinces.
... but I think these 6 are all the biggest ones (i.e. the remaining 4 are New Brunswick, Newfoundland, Nova Scotia and Saskatchewan -- or approximately all the biggest ones, because Saskatchewan is more populous than Prince Edward Island).
It also says ...
Arguably, Canada already has a 2-tier health care system because of the rigid division between medically necessary hospital and physician services (enshrined and protected in the Canada Health Act) and other kinds of goods or services for which there is significant private financing, such as drugs and home care.
... which I'm not sure is true. I'm sure it's true that e.g. dental care is difficult (barely available) for poor people, and home care too, and access to health care is sparce in small, remote communities -- but maybe drugs are available, either because of welfare or e.g. the Trillium drug plan for people whose drug costs more than 4% of income.
Anyway you might read the article if you want all its details and conclusion. It claims at the start that ...
Our survey revealed multiple layers of regulation that seem to have as their primary objective preventing the public sector from subsidizing the private sector, as opposed to rendering privately funded practice illegal.
... but I'm not sure that's so -- I think that "preventing subsidies" might be a "means or mechanism", rather than an "objective". Instead I think the objective might be as defined in the Canada Health Act i.e.
Universality: All insured residents are entitled to the same level of health care
The authors have several other articles on this and related topics for example here.