Insurance companies do. But your proposal seems to do away with them in order to lower the prices. I am trying to understand the mechanism.
The mechanism is open competition. Right now hospitals and clinics don't have to give you a price for their services, and you have to pay them whatever they ask for. It should work just like a mechanic or a lawyer, you are told the fee, you pay it, you get service. Don't like the price up front? Shop around.