Why would anyone think the Government could run healthcare?
Let's see, maybe because they already do through Medicare and other programs. Maybe because governments around the world do a highly competent job of it for better outcomes and lower cost than we incur in the US. Maybe because reflexively assuming governments are incapable of doing anything well is demonstrably false. Maybe because health insurance is a marketplace that is used by everyone and CANNOT be operated effectively or humanely without government involvement.
Is there any sign of competence or efficiency in Medicare, Medicade, or the VA?
Quite a bit actually. Not saying they don't have their flaws (they do) but they are hardly the debacles you seem to be implying.
There is not one single thing that the Government of the US has ever done more efficiently than the private sector.
What a bunch of crap. There are plenty of things the private sector does an absolutely crap job of. Policing, firefighting, military, infrastructure, medical care for at risk groups (elderly and poor especially), basic research, the judiciary, banking regulation, environmental protection, and quite a bit more. Any time you have a situation where market forces do not work well, the private sector is demonstrably unable to deal with the problem. I'm all for doing as much with the private sector as we can but the argument that the private sector is always better is absurd, wrong and frankly damaging to our society.