Those rules are completely hypothetical and borderline reductio ad absurdum, since they have never been in place anywhere in the world, and you could make the same argument about
anything the government does.
You seem to pretend that people in countries with universal healthcare don't have any agency, but of course they do. Public healthcare doesn't preclude private healthcare. We see this in Western countries that have high-quality universal healthcare, which have successfully managed to strike a balance.
There will of course always be gaps not adequately covered by the public option — new, unproven modalities that aren't offered, or unacceptably long wait lists, or a certain drug being denied that might help improve quality of life over a more conservative drug. In those situations you always have the option to seek alternative private healthcare at higher cost, and you always have the option to get private insurance. So many commenters on HN (presumably American) paint universal healthcare as some kind of draconian Big Brother regime where it's either all or nothing, and the public option will "take away my rights," when nobody has ever proposed such a thing.
In the US, you are under the thumb of private insurance companies whose profit motive is, indisputably, not aligned with patients' healthcare needs. Sure, you can shop around for insurance plans, but realistically, when faced with a health crisis, that's not an option. Which means you have to deal with a system that doesn't care about your health and tries to wriggle itself out of paying anything more than the minimum they're obligated to cover, and that minimum isn't known until the bill arrives. To my mind, having lived under both types of systems, the American scheme is much more restrictive.
Under a universal healthcare scheme, there's no profit motive to cloud the quality of care. There's a cost reduction motive that can affect quality of care, but as the other commenter points out, the democratic model helps balance that. The world over, in places like Scandinavia and the UK, funding of healthcare is a big concern that gathers a lot of public debate and figures heavily in election campaigns; it's not swept under the rug. It's not perfect, but it feels much more of a "we are all in the same boat" kind of environment than the American one where every day we have newspaper articles about huge hospital bills, health bankruptcies, drug epidemics caused by greedy pharma companies, and widening wealth inequality.