Medicare rates are based on the cost to provide services, plus allowing for hospitals and doctors to make profit. Sure, you can charge a lot more to private insurance and get away with it, that's part of the problem with our insanely high health care costs. 
A single payer system would be able to keep costs down where our private insurance system fails to do that. 
Yes, Medicare doesn't cover everything and many people get supplemental insurance to help with it. A Medicare for all system would be best if it used a Medicare type structure to cover everyone but provided coverage more similar to Medicaid.