Who's arguing that the American economy is exceptionally free market? It's certainly not centrally controlled as much as in other places, but I'm not sure "free market" is any more accurate. Large sectors (probably even a majority) of the economy is heavily regulated, monopolistic, monopsonistic, oligopolistic, etc.

The original point, that it's not fair to blame the free market for the dysfunction in the American healthcare system, is still a valid one. A main point is that "shopping around", even for commodity procedures, isn't really possible. Also, there aren't many remedies for consumers caught in undesirable contracts with health insurers. Quitting a job or moving states to fire an insurer is wholly unreasonable.