The best hospital systems in the world, like Mayo Clinic and Cleveland Clinic, are generally non-profits operating in a mixed system where they take private insurance, out-of-pocket payments, and medicare/medicaid. They have to remain competitive otherwise they won't attract customers from out-of-state or out-of-country. If you basically force patients on the public system to only use certain government-run hospitals, you're dooming them to an inferior experience than what you could get by perhaps enhancing access to public pay for some services.
1) The incentive to innovate in the overall medical field is fulfilled by the existence of private hospitals, which according to my OC wouldn't be banned or cease to exist. They would coexist alongside the public ones.
2) Behind your statement lies the necessary asumption that the medical staff would not provide a good service to their patients because the government would pay them less than private hospitals. While it is a spectrum, most decent doctors won't treat their patients worse because they are getting paid less, something that is not the patient's fault nor responsibility.
Besides, one does not choose to get ill or suddenly feel in need of medical care.
Your use of the word "customer" to refer to a medical patient sickens me.
This is one of the top hospital systems in the world, and they serve patients from all socioeconomic backgrounds. You may want to think about what information they have that you don't when it comes to delivering world-class healthcare.
We're talking about government policy here, not leadership of private hospitals.
-Private insurance is allowed to exist and compete with the public offering
-Everyone is entitled by law to the public health system for any kind of injury or health need.
-The hospitals used by the public health system are government-owned. They are run by government bureocrats and staffed with private employees (the doctors, nurses, janitorial staff, etc)
-The hospitals are managed by the respective jurisdictions in which they are, but they have to follow the basic model that the federal government imposes through the national health ministry.