Some kind of organization is going to perform the task of paying money for medical bills: figuring out which medical bills to pay and trying to get lower prices. It can be part of the state or it can a separate corporation; both kinds have historically shown the capacity to waste money.

"Middle man" isn't a dirty word. The health care deals you get in the US are much better with the middle man than without the middle man. Since the insurance companies have so much power, they can (and do) negotiate better rates with health care providers. The cost of paying for your own health care is shockingly high by comparison to the rates the insurance companies would pay on your behalf.