Better healthcare outcomes? I think you mean access: For example, the NHS looks very good until you realise that the rankings include access; Based on case-by-case outcome alone, it ranks relatively poorly - certainly less than most people would guess.
The US government spends about the same on healthcare as others do, the difference is in the amount of private healthcare spending.
The question isn't why _government_ leads to inflated cost, the question is why _American government_ leads to inflated cost.