A rarely talked about fact is that the US government already spends more per-capita on health care than European governments do. (No, really!)

So actually, it's perfectly possible for the US to enact universal healthcare and cut overall healthcare spending at the same time.

(Also, minor quibble, but when you say "Government pays for everything" you're really talking about "free at the point of use" healthcare. That's one system of universal healthcare used in Europe - and elsewhere - but not the most common one.)