There is no such thing as the free market when it comes to healthcare. Not only is the consumer not in a position to shop around when dealing with health issues, partly because you're often not in a position of even knowing what treatment you need until you've decided on a provider and undergone expensive consultations, but also, providers are put in positions of not being allowed to refuse service based on ability to pay.
Not to mention, the "free market" solution doesn't actually involve consumers "voting with their wallets" and choosing the healthcare they want, but rather choosing from a very limited set of insurance options where they have no real choice or power. And the choices they are offered basically amount to a small array of socialized healthcare options: you have no control over your benefits and extremely little ability to choose a different one, but everyone pays into a large pool and costs are shared by the group as a whole so outlier costs are not overly burdensome to individuals. It is wholly baffling to me that anyone can support this system but be against socialized healthcare. Our model is socialized healthcare, but instead of being socialized across the nation as a whole, it's socialized across a particular employer or purchasing group. Does this actually give us choice? No. Go analyze your health plan "options" offered by your employer and tell me you're participating in the free market and I will laugh in your face.
You point out the very problems that free market competition has in this market. Given that you know the problem, now try to find a solution. Hint: most of them are policy changes.
Free Market would work great if the suppliers couldn't reject people. Free Market ideas work when suppliers want to see to all possible buyers. It's grossly unfair when suppliers can pick and choose buyers based on risk profilers. Essentially, the current system is "get a serious disease, then you do not pass go".
I've seen friends get seriously dicked over because they filled out some insurance paperwork wrong. Ironically (and sadly), they would have been fine without insurance and just invested in a index fund.
I'm a huge fan of socialized medicine simply because I think it benefits everyone simply to have doctors gathering more data. All the stuff we know now is because people had crazy things happen to them (and most of them probably died), so it's in the public interest that people don't go off into the corner and die; rather, the health data is valuable (especially for the rich and upper-middle class who I presume would like to live longer).
Eliminate drug patents, remove the annual cap on medical students in doctoral programs, remove that requirement entirely from a medical license and let anyone practice medicine that wants to (albeit with some regulation requiring a practitioner explicitly state their qualifications, and you would have to prosecute impostors).
That would probably lead to what we have today - people only trust literal doctorates from top medical schools - but you could open new medical schools to placate demand, and hopefully reduce the workload required to obtain a medical license (the current 8+ years of schooling is excessive for any profession) or provide some kind of doctoral certification program outside the bounds of academia, though I have no intimate knowledge to know how you would do it.
It does become buyer beware, but it also means you would have treatment at most pay grades. Also, if it wasn't illegal to produce chemical compounds (really, that is dumb, and all the arguments of how they produce progress because of profit motive is pointless if nobody can afford the result, and the investment funds in big pharma would find other avenues to stimulate growth and research). The costs of care would fall absurdly fast if there weren't a dozen golden gates on the road to being a medical professional or medical facility, I wouldn't even mind if they just reduced the barriers, but they are too high and too plentiful right now.
Because if I can take from you without consequence, then that amounts to a great level of control. Normally if I take something of yours without your consent, I am a thief. If I do it at gunpoint, I'm an armed robber. Take away the veneer of legitimacy that many would grant to the DEA, and it, too, becomes a thief and/or armed robber.
