Emergency services aren't a good fit for markets. For markets to work, people really need the luxury to shop around with some leisure.
Wasn't there some Roman oligarch that ran a private fire department, and basically just used it to extort property owners when they had a fire? Truly market based emergency medical care without legal restrictions would likely frequently resemble that with some frequency.
Edit: Yep: https://en.wikipedia.org/wiki/History_of_firefighting#Rome
> The first ever Roman fire brigade was created by Marcus Licinius Crassus. He took advantage of the fact that Rome had no fire department, by creating his own brigade—500 men strong—which rushed to burning buildings at the first cry of alarm. Upon arriving at the scene, however, the firefighters did nothing while Crassus offered to buy the burning building from the distressed property owner, at a miserable price. If the owner agreed to sell the property, his men would put out the fire, if the owner refused, then they would simply let the structure burn to the ground.
And that's the open market; it gets even more cloudy when we talk about employers being involved, so we have an agency problem as well.
It's a system unlike any other, and I don't think general theories really apply very well as such.
Hospitals increase rates to cover the overhead of dealing with hostile insurance companies. Insurance companies raise rates to match increased hospital billing. Ironically both industries are incentivized to raise rates on customers while trying to gouge each other.
Is it any wonder healthcare is a scam?