Clearly, that is not the case.
And in regards to your last point, what has made the system so perverse is the insurance companies in the first place. If most people were actually responsible for paying all of their healthcare costs themselves, the prices would be pretty cheap.
For example, I live in Tanzania, and I can get medication that is over $500/month without insurance in the US, or like $75 with insurance, for around $1-$2. Also, I can get a tooth pulled for under $5.
No one has health insurance here, I don't think it even exists, and therefore, hospitals are forced to charge an amount that the people can pay for. The system actually works, and if for some reason you can't pay, you can volunteer at the hospital until your bill is paid off.
Consequentially, even though there aren't laws about accepting all patients (hospitals are allowed to turn away whoever they like), they don't, because they know they will be paid at the very least in labor.
Though, honestly, at this point in the US the system is so fucked up that I'm not sure it's even possible to create a free-market system anymore. It pains me to say this (as a staunch libertarian), but I think maybe it would be better at this point for the government to just foot the bill for everyone.