The US pays vastly more to administer it's health care than any other OECD country. Your assertion that the cost of government bureaucracy should exceed the cost of private markets at this task is emphatically not backed by data.

A good place to check your assumptions on these kind of basic things is the OECD data. Healthcare is here:

USA 1.4% of GDP on governance of healthcare Germany 0.5% (competing public insurance schemes) UK 0.2% (single payer)

US style healthcare is an ideologically driven luxury.