Some employers also offer as a bonus a sort of subscription at a private clinic, so you can see a private doctor or have an operation for a lower price or even for free.
In the USA the government health programs for people in low incomes, children and pensioners cost about as much as a typical European single payer health system. Then tax payers get to pay to be gouged by health insurance companies to get any cover for themselves.
If any regulation at all makes a market not "free", then there are no free markets as soon as we have any laws.
Like all free markets, this one is regulated. There are degrees of freedom.
This is why this isn't a free market. It's not about regulation, it's about the system being divorced from responding to market dynamics.
Aside all the insurance stuff, you cannot open an MRI imaging lab or similar without a letter of need from the local government. The supply side is quite literally gated by existing players in the market (via campaign bribes and similar).