Going one step further, just eliminate insurance, and the entire billing system becomes unnecessary.

My mom had an experience in a country with national health care. She got injured while hiking, and got treated at the clinic in the first town she reached. Whey they were done, she asked how to pay. They said: "You don't pay for health care." Well, what about foreigners? It turned out they simply didn't have any way of figuring out a price or taking her money. Those mechanisms didn't exist. It was probably cheaper for them to dish out free care to a few foreigners, than to manage a complete billing and collection infrastructure.