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.
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.