I don't think it would work well. Hospitals are not a free market in most of the US. You have to get a Certificate of Need to set up a new one. Given the near monopoly they have in the local area, regulation is probably going to be required if nobody gets to negotiate.
Why can't a hospital charge the same price for the same procedure? I can see that different hospitals may charge different prices from each other. But a single hospital should charge the same no matter if uninsured, insured or whatever.
What is to stop the local hospital from deciding that an appendectomy costs a million dollars? You pay, insurance pays, no matter, is is a million bucks.
The only business that works that way is retail. And even then most will negotiate on bulk orders. Almost everyone else will have a price list they negotiate off of.