I'm abandoning my libertarian beliefs a bit, but you are kidding yourself if you think there are "problems" with health care in other European nations. Besides the immense cost to the government, people in Germany, Denmark, etc get a superb product. So not, unless you take into account people who work for private insurance companies, there is no "trade-off" taking place.
I am looking at the chart in the thread here, whether it is accurate I don't know but it makes sense based on what I know already, at least in the case of the US.
Where some places have both high quality and good access, it is likely because they have worked through the problem in a way that has found an adequate solution, not that a potential for such a problem does not exist. You were just referring to scarcity after all.