Any system implemented in the US would almost certainly retain private doctors and hospitals, even if gov’t would be the benefits admin. Zero people are proposing an expansion of the VA system to cover all Americans.
We’d probably end up with something like Medicare for all legal residents + Medigap private insurance. If virtually everyone has it, then most doctors will have to accept it or lose business.
You’d end up with basic coverage portable between jobs and states, with highly predictable costs. Great for worker mobility, which is generally good for an economy. Maybe not so good for bad employers who use insurance as their only way to retain workers, but so what?