That seems pretty reasonable. Private healthcare has done the same thing as private education here, allow certain people access to incredible luxury while denying others what should be basic rights.
Yeah, and I know none of you is a politician (well, I think), but this would be an idea at how to not alienate those who also want private schools/healthcare/whatever to thrive beside public institutions or at least as an option:
Set a high baseline of standards for everyone and tell the companies that if they want to stay in market, you need to one-up these standards.
And I think that should work with pretty much everything, even when something gets privatized. Privatizing the water supply for instance wouldn't have been so disastrous if they had set high standards that had needed to be followed else the contract would be automatically terminated. Of course, those might have shooed out the buyers as then there wouldn't have been much of a way to profit from it anymore...
It should be that way especially when something is privatized. With health insurance however, I'd prefer that with a national universal program like Medicare that all other comparable products be outlawed. The reason being that the strength of such a system comes from creating a monopsony, all the American people buying healthcare as one, thus giving maximum bargaining power to the people. In general, I'm skeptical of the very concept of health insurance. Health care is a real service, its supplies are real products. For-profit health insurance is not a service, it is rent seeking. A gatekeeper on services, nothing more than a worthless, unnecessary middleman, a burden on the system. To allow such a parasite to gatekeep a basic human right like healthcare is immoral. If there are luxury services beyond what a comprehensive healthcare program would provide to all, and a significant enough group of people have the means to buy those services so as to support a market outside of the basic services guaranteed by the government, but was not so affordable that insurance wasn't required, then and only then would I allow for private health insurance. With education and other public goods I can see the system you suggest here working, and indeed even with healthcare such a system might work (see the UK's system) but with health insurance, I have stronger opinions. Private health insurance may sometimes be feasible and affordable when done correctly, but I nonetheless consider it fundamentally wrong, and only tolerated by society because it has become normalized.