Just ask any veterans relying on the VA how satisfied they are. On a tangent, universal health care is not 'great' anywhere, and the bigger the country, the less compelling the evidence (diseconomies of scale -- see Canada/England vs. Switzerland for example). And our country benefits from another order of magnitude of diseconomies of scale/bureaucratic bloat in trying to institute such a program.
So, moral of the story is that if it somehow did not end up making our healthcare system significantly better than universal healthcare countries, then we could still let states that lean a certain direction handle the bureaucracy of a universal healthcare system with slightly more efficiency.
I don't think there is. I think in many cases, what starts out as a "free market" is ultimately overtaken by a few large corporations (as others fold or get acquired) which then proceed to abuse their position of dominance to engage in monopolistic rent-seeking behavior to maximize their profits at the expense of everyone else. Think Microsoft abusing their dominance of Windows, or Intel abusing their position to prevent PC manufacturers from using AMD products. We're starting to get there with iOS and Android being the only two remaining choices for mobile, and Facebook for social.
My concern is even if you could hit a magic reset switch and turn the health care industry into a free market, it would trend the same way. Insurance companies would grow in size and squeeze the health care industry from both ends, charging patients more and more money, while paying doctors less and less, all so they can squeeze out a larger profit. Pharmaceutical companies would jack up the prices of their drugs because they have a captive audience, they know you're going to pay up because your life is at stake, all so they can squeeze out a larger profit.
It's questionable to me if the medical system could ever act like a free market:
1. In emergency situations you often don't have the opportunity to "shop around" for care.
2. If a pharmaceutical company is the sole manufacturer of a life-saving drug, what stops them from jacking up the price of said drug to maximize their profits, at the expense of patient lives?
3. If paying for someone's medical treatment is unprofitable, insurance companies have an incentive to drop that person's coverage instead of paying for treatment. (And often did pre-ACA.) How could a free market possibly address this situation?
Universal health care systems around the world are not perfect, but many countries have implemented them successfully and as a result spend less money on health care (as a percentage of GDP) than the US while simultaneously having better health outcomes. Why not move towards the known quantity which has already been tested and proven by every other modern country in the world, instead of this hypothetical free market system which we've never seen before, and frankly, sounds rather dubious in light of my questions above (especially (3))?
So, do you think everything's already been solved? There have been 100 billion people in Earth's history. Wouldn't you think one of them would have proved P vs NP by now?
If you have a legitimate argument against a cleaner, free(r), less corrupt health-care system, feel free to bring it forward.
But no, in modern medicine's one century of existence, the problem of finding the best health-care implementation has almost certainly not been solved. In fact, only a dozen or so countries have had the economic capability to sustain modern medical practice on a large scale for more than a couple decades. The others constitute the "second/third worlds."