The medical system is really not private. It is a hybrid of a state-controlled system and a capitalistic system. Here are some ways of how the system is state-controlled:
1. Hospitals cannot be freely constructed. One must apply for a "certificate of need" to gain permission from the government to build a hospital. This is not a free market.
2. Credentialism is BAD in our medical system. There is only one group of people approved by the government to do prescribe most medications and perform most surgeries - doctors. This is not a free market and greatly increase costs. Doctors earn a median of 300k/year and are the highest paid profession in every state. This is not a free market.
3. Hospitals are legally forced to treat anyone who shows up at their door - even if they can't afford the treatment. This is not a free market.
4. Medicare/Medicaid are forced to nearly buy any drug, regardless of cost, if it could save a live. Even if a drug is 500k/year, the system is forced to buy it. This is not a free market and greatly increases costs.