However I'm willing to bet that the people who are actually paying for the system would face threats of violence if they stopped paying.
We do have private health insurance here in the UK, but I'm sure there are restrictions on who can do business and where and when. I think that is unavoidable. There are obvious difficult trade-offs when legislating this kind of thing. Not everyone can be made happy. However, the freedom to operate a medical business without government interference is not as important as ensuring that everyone has access to adequate healthcare. If your rights as a business owner get trampled on at the cost of saving human lives, well, frankly that's just too bad. There are plenty of other lines of work you can get into where there aren't quite so many dangerous and perverse monetary incentives.