The US has ~330 million people. The US has ~13 million undocumented immigrants. The US is the size of Europe. You're not going to magically get the efficiency of European countries 1/10 the size by having single-payer/universal health care.
Imagine trying to have a single health care system for all of Europe. It would not be nearly as efficient as you think it would be.
Does that mean we can't significantly bring down the cost of care in the US? No, we absolutely can. But a national system in the US is never going to be as efficient as any European country's national system. It's impossible to maintain the same level of efficiency when the scale increases exponentially.