Patients in distress are usually not very able to make distinctions between hospitals based on pricing or really any other factor apart from perhaps proximity. This key point seems to not be addressed by any of the standard libertarian takes.

Edit: Also, I grew up in a country (India) where healthcare was definitely part of the market economy. And what that meant (and still does) was that people with money get high quality immediate care, and those without get shafted. That's what the market does to healthcare.