The problem here is one of incentives. If I promise to pay you for mowing your neighbor's lawn, and he rarely reviews what you're doing and I rarely check with him, suddenly mowing the lawn gets really complicated and expensive. If, on the other hand, you walk next door and solicit the neighbor to mow his lawn, suddenly it's a very simple transaction. The difference is one of incentives. If the goal is to provide mowed lawns, the neighbor is the guy responsible. If the goal is to provide a rationale for writing a check? Heck, I got all kinds of easy rationales handy. I can play this game all day.
It should be illegal for anybody to pay a medical bill except for the person receiving treatment. If you want universal coverage, that's fine: pay the patient and have the patient pay the hospital and doctor. Once you begin to incentivize other players besides the patient, you're going to create a mess. I'd also recommend completely unbundling hospital services. Patients should be able to buy materials and services on the open market -- and keep the savings they make doing so. Hospitals should provide clean, sterile rooms for independent specialists to work, not form a pricing cabal controlling anything that happens inside the walls and incentivized to game the system.