Yes, there are problems with socialized medicine. But there's problems with US style medicine too. Theorists can argue forever about this, but it's pretty empirically clear that the US system sucks, and costs a lot of money. The other systems suck less, and cost less. Which would you prefer?
I didn't say anything about US medicine, especially whether it has price problems and whether it could or does have price control problems. I only gave a specific answer to what happens when there is inelastic demand because of a government that can also regulate the suppliers. I see how "this is known" might be taken the wrong way, but all I meant by it was that the OP didn't need to wonder...it's found in any standard microecon textbook.
I would like to think we could talk about cause and effect in isolation here without making everything a grand political statement.