According to what I understand, the U.S. has had three chances at something approaching universal healthcare—in the late 60s/early 70s around the same time that Canada was dancing around the same question; in the 90s under Clinton; and under Obama. Given that it took about fifty years after Canada[1] to even get this passed…it will likely be a long time before anything rational gets passed.
[1] Per Wikipedia, the Federal framework for jointly-funded universal healthcare in Canada was passed in 1966 under Pearson, and it was strengthened in 1984. OHIP was first created (as OMSIP) in 1966, reinvented in 1969 (as OHSIP), and finally renamed in 1972 to OHIP.
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.)