"Seems obviously a problem of size. Can you point to a working healthcare system in any other country that has a population anywhere close to the US?"
Healthcare in the US is run at the state level, moreover, in Canada it's also run at the Provincial level.
Germany's healthcare 'system' is bigger than NY and California combined.
In Switzerland, insurance is at the cantonal level.
Scale is not the issue - in fact, scale should mean lower prices.
There actually is not a lot of competition for Medical services in the US, you have very powerful guilds with power handed to them by the gov.
It's almost Mercantilist in a way.