One major oversight that you fail to address in you hypothetical free market scenario is that supply is still constrained. In a truly free market, anyone can set up shop and try to market and sell their services. If I buy a TV that craps out in two days, I either get it fixed by warranty or I go online and write a bad review which hopefully contributes to the demise of the company. If I get a bad surgery from a poorly trained doctor I die. That is why we have the FDA etc, and that is why doctors are rigorously vetted and trained through the med school/residency process, and I think most people would agree that that is a good thing.
Another issue I have is your mention of 'charity'. Almost anytime this gets used in a political/economic discussion, it is used to dismiss a whole class of problems that the author does not want to deal with in their argument. In this particular case you mention that if you cannot afford a 'free market' doctor/treatment, then you can turn to charity to provide the funds. Of course we all know that charity will never come close to providing for the entirety of society, so you are left with a situation where almost everyone except those who can afford treatment are forced to go without it. And of course the typical libertarian response is "tough luck" or "work harder". But here is the thing, health, similar to environmental issues has strong public effects. There are a number of diseases and illnesses that, if left untreated in a large portion of the population, can have devastating impacts on society. A healthy population is a necessity of a first world economy and relying on charity to ensure positive, society level outcomes is absurd and ignores reality.
Look, I lean libertarian on many issues, but there are certain aspects of society that are just not suited to a free market. I want to be able to go to a restaurant without being terrified of catching TB from the waiter or getting salmonella poisoning from the food. I don't want industrial companies polluting the ocean and rivers that I surf and fish in. The free market can be very powerful, but it is also fairly dumb, it cannot recognize when it is doing more collective harm than good. I don't see why so many libertarians think in terms of all or nothing. Compromise shouldn't be such a dirty word in today's political discussions, it's a sad state of affairs.