People with Medicare or private insurance simply don't care what the bills come to because it doesn't come out of their own pocket. There is an almost total lack of consumerism in the health care market. This is what drives prices up. If we passed a law forbidding health care providers from charging different prices based on insurance coverage, we would even the playing field for consumers to make better choices. How much insurance do you need? How much is applied to a deductible? What can I expect if I have to spend a night in the hospital? What's this going to cost?
This wouldn't set government prices on health care providers, it just makes them charge everyone the same.