But then the best facilities and the best doctors would all be in private practice where they could earn more money, and the public care would be done by whoever was left over.

I think that’s a bit of an exaggeration. Most healthcare workers got into the field for reasons other than money. And with a well-funded public healthcare system, I think the US could still support pretty good wages. For example, the VA and military can attract a pretty good standard of doctor, partially through the offer of paying for medical school and with excellent benefits. I don’t think that every top specialist will opt to serve the rich in private hospitals, or that there would even be adequate demand from those rich to require the supply of every top doctor.

That actually depends on implementation: there are countries where private practices practically don't exists besides people in the 1% who pay for the convenience and attention; and countries like Poland where a lot of people have work-provided healthcare benefits which mostly decrease wait time for specialists - very limited subset of healthcare overall, there are no common private cancer treatments etc.

I don't think there is country where people buy very broad private coverage after having mandatory public one, so the scenario you're trying to present wouldn't happen: there would not be enough jobs for "all the best doctors" in private sector.

In the UK system its pretty common for senior doctors to work for both the socialized NHS and private hospitals.