That also does not exist in the US and hasn't in more than half a century. The US could hardly be any further away from a free market in healthcare. It's hyper regulated, and hyper dominated by government programs (Medicare, Medicaid, and dozens of other programs and agencies).
The US healthcare system is controlled top to bottom by rigid, strict government regulations and oversight. It's the way it is precisely because there is no free market at all. The corporatist interests like it just the way it is, and work with the politicians to constantly maintain that highly regulated environment: it's regulated in their favor, exclusionary of nearly all potential competition. And it's insanely expensive to try to start anything in healthcare, which favors the incumbent further.
Try starting a hospital or opening a clinic. Try becoming a doctor. Try getting a drug to market. Try getting a medtech device to market. Hyper regulation every direction you look.
Free market? Ha.
We don’t really know what a true free market would do with modern healthcare because as you say there is no such thing. Personally I think it could work for middle class and above but would leave a big coverage cliff down below because it’s not profitable to insure or treat the really poor.
The question is whether real transparency with price competition and efficiency gains would make it cheaper to then add a safely net for the poor.
There are no free market health systems in leading economic nations (unless maybe you take a very small niche corner of the systems)