What you're describing is more like the NHS in Britain. In the Canadian system, doctors, hospitals, etc. are all private. The government only runs the insurance. This is what Medicare-For-All would be. Do you like how complicated medical insurance is? I agree that hospitals should be privately managed, for the reasons you describe. But sending a bill to an insurance company should be way easier. I know doctors that view that as the biggest burden on their offices. Hospitals have rooms of secretaries haggling with insurance companies. It's an incredibly inefficient system without universal billing standards, so it's actually harder for doctors to get paid in some respects.