Nosewelder
Some subsets of the health care industry have price elasticity and seem to work fine in the free market, such as Lasik and liposuction etc. But stuff that gets called medically necessary don't work in a market, there is an entire branch of economics dedicated to it: Health Care Economics. Ironically pioneered by an American, Kenneth Arrow. Won him a Nobel Prize.
The difference in payment between US and European health care workers isn't large enough to make much of a dent in US healthcare expenses. The big sources of costs is the US massive bureaucracy dealing with bills, insurance and negotiation, as well as the medical inefficiency and overprovision inherent in the system. Also high drug prices.
For example, a US medical doctor makes about 100 000 $ more than the European average, although a few countries pays doctors more. There are about 1 million doctors working in the US. So thats 100 billion. US overspending is about 1 500 - 1 800 billion.