> Also, Medicare/Medicaid are cheaper primarily because they pay providers a lot less than private insurers for the same service.

Do you have numbers to back this up? I'm not saying you're wrong, but my mom worked for a GP for ~20 years (recently retired) and listening to her talk about it this was not true for her employer at all. She said that in many cases the private insurer paid less, and on top of that required more back and forth to get claims addressed.