There are, however, no examples of successful health care based on the
principles of the free market, for one simple reason: in health care,
the free market just doesn’t work.
That's Paul Krugman. I would frame this point a little differently. There are in fact plenty of people who buy their health care in a more or less free market setting, most of all in Latin America but all over the world. It's far from obvious that these markets fail in efficiency terms ("compared to what?" is the obvious follow-up). For the wealthy in Latin America these markets seem to work well. They work much less well for the poor but is that because of market failure or because these poor simply don't have much money to spend?
One possibility is that the main problem with these markets is distributional rather than efficiency. (Krugman's third paragraph recognizes this, but he doesn't use the point to reorganize the analytics of his critique. Also note some tricks. When markets fail at providing insurance, ex ante this is a possible efficiency problem but ex post it will be a problem of distribution.)
Another way to state the health care problem is this: once we try to obtain distributional objectives, supply becomes less efficient. That understanding might focus your attention on a voucher-like system, combined with deregulation. rather than government interference in provision. Another option is for government to provide nudges to have better monitoring of HMOs or insurers, to make them more trustworthy. Or maybe catastrophic-only insurance, to overcome the distributional problem where it is most severe.
You can understand the French system by citing the incentive for overtreatment and now we are back to the possibility of efficiency being the primary problem. If you limit overtreatment, by organizing doctors into poorly paid, fixed salary co-ops, you keep costs down and make some parts of the distribution problem easier to solve.
There are plenty of health care services in this country, such as laser
eye surgery, or plastic surgery, which are supplied in more or less
market settings. I don't consider their efficiency an open and shut case, but it's quite possible we'd be delighted if other areas of health care worked this well in terms of cost-lowering and innovation and even availability. It could be that these services are more transparent or it could be they are simply less regulated and further removed from third-party payment.
Read this post of Bryan Caplan's and ask yourself whether the Arrow problems are in fact what motivate most of the health care intervention we observe in the U.S. Maybe France is the country which took Arrow seriously.
It makes a difference whether you view the case against the market as starting with issues of efficiency or distribution and usually those concepts are jumbled together.
Sometimes I wonder how wealthy we all would have to be before we could just pay cash for our health care. I call this the Pablo Escobar solution. It's a long way away but is it imaginable at all? Does it recede as we approach it? Do we have to give up some distributional objectives to ever get there? Do we simply embrace it when the poverty line is defined as standing at $200,000 a year?