The problem with that is that it requires not treating people who are uninsured and unable to pay for care. "Sorry you got hit by a drunk driver, but we think that the risk is too great that you and the guy who hit you won't ever pay the $100,000 it would cost to fix you." That's pretty unpalatable.
Also people with pre-existing expensive conditions would need to pay a lot more.
Either, because insurance companies charge them more.
Or, if insurance companies are not allowed to charge them a special price, healthy people will be reluctant to pay for (now) overpriced (for them) insurance.
One clever technical way out is to have your parents buy insurance for you before you are even conceived. This way, because neither the buyer nor the seller of insurance knows what's coming, it's not a market for lemons. (And if the parents genes make this kind of insurance too expensive, because it's expected that you inherit some defects, perhaps they should rethink their decision to procreate with each other. (Just like couples with sickle-cell-anemia on both sides are already advised.))
But of course, people will not be that farsighted, and we don't want to penalize people for their parents making stupid decisions, like not to buy insurance, more than necessary.
So in actual life, a basic version of an NHS like system plus optional extra insurance you can buy seems like the most sensible policy.
I see what you're saying, but how can one get a market forces involved so that the price of care isnt super high and that users have some choice.
I have lived in a country with universal healthcare. Its not that cheap, my portion of taxes that went to healthcare was about the same as the highest cost Kaiser plan. But Kaiser (thus far) is leaps and bounds better than the care received.
Where I am from you make an appointment with a doctor, and they will almost certainly be 30-60 minutes late for it. Then when you do get in the room with them, they will give you 15 minutes max because thats all the government pays for. Similarly every person I know who needed something done is on a waiting list. For months and a small number for years.
People complain about the cost of US healthcare, but if you can pay for it then it does seem excellent.