I don't think many economists would claim that free markets offer a perfect solution regarding health care. Rather, they would argue that the alternatives (for example, a government run system) are worse.
It's not about government vs. private market, but about whether a society is willing to pay in solidarity with their peers for the less often but expensive and perhaps complicated procedures trough taxation.
Or whether they think that it is better to have a system where those who can afford get the best treatment and pay through their insurance.
IMHO a modern society finds a balance between what to lift collectively and what to make a personal responsibility. Not because it is moral just, but because it is financially a more useful model as long as it is to expensive to run through the private system.
But again, it's not black and white, there are many arguments for and against.
We get better health care in the US.
The US also subsidizes drug discovery for the rest of the world. In the late 90's, I worked for a company that made robots used for drug discovery. Every once in a while, they would put up a chart showing year over year growth. There was a healthy, steady climb until the Hillary health care initiative. The company almost went out of business. Drug companies froze spending on drug discovery because it wasn't worth their while to spend the hundreds of millions to bring a new drug to market if the US was going to go to a single-payer system. US drug prices pay for R&D, and all the countries with enforced pricing get a free ride.
Is it because you think the public plan would negotiate drug rates too low?
Don't the major private insurance plans already negotiate drug rates?
It is true that America is an innovator in health care, but I don't think that will change when the govt is signing the checks.
We also have the best missiles, bombs, and tanks. And all of that is govt funded.
You conclusion doesn't follow from the data. The only conclusion you could make from that data is that the US has better care for prostate and breast cancer than Europe.
Most people don't die of these diseases. Indeed, most people don't die of cancer. So we should be looking at survival rates for all conditions, i.e. life expectancy.
As it happens, most (all?) western European countries have longer life expectancies than the USA, and all spend less on healthcare. If you wish to conclude that the US system is the best one, your conclusion isn't supported by the facts.
The average life expectancy in the EU is 78.7 and in the USA it's 78.06, according to Wikipedia.
In 2004 and 2007 12 countries from central and eastern Europe joined the EU, with lower standards of living and (presumably) healthcare. The western EU countries are more comparable with the USA, since they have a more similar standard of living.
If you compare the complex mess in the US to a simpler federal-based system, of course the simpler one looks better. However, remove all regulations and tax implications from the US and prices will dive bomb and quality will improve.