This is a great question. My understanding of it is as follows:

There are legal requirements that force doctors and hospitals to offer only the state of the art care. A doctor who fails to do this can be sued for malpractice.

Imagine if a homeless person entered the ER and based on his perceived ability to pay doctors tried only $1K worth of interventions before giving up. This would be viewed as inhumane.

Most healthcare is paid for via health "insurance" which is heavily regulated. Via medicaid and medicare, the Federal government establishes baseline market prices for different services. This is done with a broad, public health perspective in mind, with the idea that the market for preventative care (largely primary care) should be larger than it would otherwise be, because treatments are generally much more cost-effective when problems are identified sooner and treatments initiated sooner.

It's important to note that the free market would not likely offer highly skilled primary care docs for most people, since the price would be prohibitive. Instead most people would see a less-trained person such as a nurse (or an NP or PA). This already happens today, but the liability loophole is solved by a process where the MD signs off on medical decision making.

When it comes to primary care, it's important to point out that the body is largely self-healing and many illnesses simply go away with no treatment. Often, treatment outcomes do no better than a placebo.

Due to this reality, the free market would lower the caliber of care available to most people who are not extremely ill, and without constraints on who may practice medicine we'd see all sorts of alternative therapies and unscientific therapeutic approaches elevated to the same status as those that are scientifically backed. The market would have no way of discerning the placebo effect from actual measurable effects (and it perhaps shouldn't matter).

For seriously ill patients, the quality of doctors matters immensely, but a free market system would certainly reduce the availability of highly competent practitioners, since those who need treatment for serious illnesses are not always able to pay for them, and few people anticipate needing them, so insurance premiums would increase due to lack of demand and a smaller, adversely selected risk pool.

In addition, for seriously ill patients in the US, a great deal of money is spent on end of life that has no public health benefit. We overpay for healthcare largely for the privilege of having our loved ones put through extreme end-of-life measures just in case another year of life can be obtained. France and Canada do not do this, and their cultural approach to end of life is better in tune with the probabilistic outcome of end of life interventions.

Put another way, in the US we consider it a greatly important volitional act to have every possible measure undertaken to keep us alive. Insurers cannot offer a lower tier plan that (say) mimics France's approach because of liability concerns.

It's not possible to imagine a free market for end of life care without a stance on the liability landscape for care that is anything less than the most aggressive (and costly) intervention currently known.

So what do we get in exchange for this convoluted system? Our doctors are selected out of a highly competitive group of motivated students and trained extensively. Most are overqualified for the vast majority of the work they do, but patients prefer the Cadillac doctor over the one who may seem only marginally qualified.

Thus we revere doctors and treatment seeks both a cure and a sanctioned course of action from a figure holding social authority. This phenomenon offers a fascinating look into the psycho-social aspect of illness and the need that patients have for validation of their plight from an authority figure. Just as a doctor's note is an excuse for missing work, a doctor's expert opinion offers validation for the sick in a society that prizes youth and vigor.

So we pay a premium both for the extra (usually unnecessary) training and also for authority. We get (in a sense) both a treatment and a blessing.

It is a uniquely American phenomenon, I believe. The hope that we feel when a surgeon agrees to do a risky surgery on our 90 year old family member is the same hope that we feel when spending our last quarter in a slot machine or risking everything to do an unlikely startup.

What does this sort of uniquely American healthcare look like in a totally free market system? I'm not sure. But I think it would more closely resemble a casino or a church. There would be more emphasis on hope, more emphasis upon meaning, more emphasis upon the authority figure aspect of healers of all kinds (not just trained physicians).