You can argue that single-payer is not the solution, that's valid. We can discuss that. Throwing 'free market' is also valid, except that Medicare/Medicaid didn't exist forever.
We already had a 'free market' system before Medicaid and Medicare (and even Social Security), and Medicaid/Medicare/SS were actually invented to solve actual problems that the free market couldn't tackle.
That's because the healthcare market, much like the retirement market, isn't your average economic market. At some point, the people in that market stop earning money, and depending on how they did in life, or what kind of events happened throughout their life (through fault of their own or not), they ended up with more or less money, and thus more or less access to care or anything else.
I love the free market, and I think it makes sense in almost every scenario where you have a relatively stable market. When it's not stable, you end up with access problems (that is, only the section of the market that's profitable gets the service). This is common sense capitalism, and why the government had to intervene in a bunch of markets to ensure access (another good example of this is the Universal Service fund in telecommunications).
At some point there's a state interest in the people having access to some of these services. That sometimes conflicts with the profit motive. It's a real issue that, especially on healthcare, simply has to be addressed by whatever solution.