If you could magically make healthcare look like an Econ 101 market where rational choice theory and perfect competition both were even remotely reasonable models of what was going on, sure.
But "patients don't pay directly for services" isn't even close to the only way that healthcare doesn't look like an Econ 101 market. Rational choice theory is a deeply and fundamentally flawed model to start with, but its a tolerable approximation of behavior in lots of important cases (mostly, classes of products which are frequently purchased and used and whose utilities are very clear upon use, allowing any misperceptions about expected utility to be resolved so that most purchase decisions are in fact made with reasonably strong information about costs and utilities.)
Look, the rest of the developed world does as good or better (by most measures) healthcare than the US at vastly lower cost (whether measured per capita or per GDP), and they don't do it by making patient pay directly for services. The whole idea that we have to resort to abstract Econ 101 theory to determine how to control prices in healthcare ignores the fact that the rest of the developed world exists.
The theoretical efficiency of a real "real free market" depends on the ability to evaluate and choose options in a competitive market for purchasing goods and services based on perfect knowledge of the costs of and utility from each purchase.
There's a whole lot of reasons that a deregulated market for "purchasing treatment for a heart attack you are currently experiencing" is not likely to have the features that theoretically create efficiency in a "real free market".
You can't research the market and just skip getting a knee replacement altogether, the way you could with lazik. To say nothing of more serious problems.
And there are the other couple of billion people on the planet that cannot afford a dollar per day for a meal.
...partly because of healthcare costs, which is the number one cause of bankruptcy in the US. You're making our point.
OTOH if I get in an accident where I'm not at fault (let's assume the other person didn't have insurance), I'll use my insurance to fix the car, because it's a sudden emergency with a high cost. I don't see how that's different than a heart attack or whatever.
I don't see why people conflate all medical care into one group. Routine care and emergency care are clearly separate.