It will be a disadvantage for the free public health care to have a monopoly on health services.
How unfortunate, then, that it effectively would. The vast majority of people will have no other choice but to use the "free" public healthcare, as private doctors, if any exist, will be exceedingly expensive, and of course, charge for their services above and beyond the already nearly crushing tax burden required to fund the "free" healthcare.
Reality check: Do you have a free public school system? Are there private schools as well, or do the public school system have a monopoly on running schools? If there is demand for better or different services than the public provides, there is a market.
Indeed we do. And the case that I explained for private doctors is precisely the situation which exists for private schools. They are quite expensive, and of course sending your kid to one does not get you a tax break. And I did not even touch upon the quality of the service provided: Public schooling is widely acknowledged as being sub-par, but while everyone knows that private schooling is much better, only the wealthy can afford it.
Countries with a free public healthcare system spends half as much taxpayer money per capita for a much more effective system than there is in the USA. Your "nearly crushing tax burden required" is half of what you pay for the system now.
That may be true. Certainly this half-public, half-private monstrosity needs to die. But of course, those other countries are
not the US, and therefore have different demographics, and a smaller population, and are therefore a smaller burden on a healthcare system.