The key point remains that the US govt has a huge role in the current healthcare system. The programs that it runs that have comparable scope to the private programs do not produce better results or cost less.
Why should anyone believe that they'd do better if they covered everyone?
That's not the key point of your argument at all. Your thesis is that everyone is covered, under different programs. That's demonstrably, decisively false: the "program" that covers the uninsured first denies them long-term preventative coverage, then forces them into emergency rooms, and finally drives them into bankruptcy. It is the most expensive, least efficient mechanism for all parties involved.
The easiest way to come to the conclusion that the government would do better is to recognize that the only way we could possibly do worse would be to let people die in the emergency room instead of treating them.
Your argument appears merely to redefine the word "covered" to mean "screwed".
Why should anyone believe that they'd do better if they covered everyone?