"Bad" emergency hospitals would eventually be blacklisted by medical insurances. Your ambulance would probably work to/with your medical insurance, and would not take you to the blacklisted hospitals. Or at least avoid it if possible. That would of course create a strong incentive for hospitals not to be blacklisted, since being blacklisted by most insurances could drive them to bankruptcy. And that would ensure "bad hospitals" to be rare and of quick living.
By the way, the assumption that you always have to go to the nearest hospital is exaggerated.
Under these circumstances, there can't be a market solution.
He, it's funny how fast people conclude what they want to conclude. It took me only a couple minutes to come up with the answer above. And I'm just one random guy. How dare you say millions of people working freely wouldn't be able to come up with a much better solution that we cannot even imagine today?
It really looks like what people against slavery abolition in the 19th century would say: "But who would do the hard work? Agriculture would collapse! We would all starve! Do you want to live in a society where people starve to death?"