The health systems in Europe are not radically different from the US system. The efficiencies of each are difficult to quantify without context. For example, the US invents most of the drugs and medical technology used by the world. Would this still happen if there was more invasive regulation? We can't know.
Besides all of this, think of every other area of the market where the government insinuates itself. Are public schools better than private schools? Almost never. Is the US postal system better than FedEx? Of course not. The government is not a commercial entity. The incentives and influences on it are not conducive to producing quality products at good prices.
"Rights" are basically just freedoms we've decided we like a whole lot and want the government to guarantee, so that's not exactly a static category. So on the one hand we can just decide whichever rights are being violated are—at least conditionally—not rights. Boom, done.
On the other hand, is there any such thing as a government that does not force people to do things they wouldn't voluntarily do? I'm pretty sure if you take that away what's left has lost the single most important trait of a government, and what's left is... I don't know, a non-profit begging for money on TV and utterly incapable of fulfilling any of the usual roles of a government? In that case, the fact that single-payer would force people to do things doesn't per se remove it from the realm of legitimate government activity.
As for the utility of the term "single payer", I believe it's distinct enough from the much broader "socialized medicine", which could include things like seizing hospitals and making doctors exclusively state employees, to make it substantially more than just a euphemism.
How is that not the world we live in? What is the alternative? At some point someone's deciding what's a right and what isn't. Other people may disagree. Rights aren't a law of nature, they're a fuzzy category, subject to conditions and caveats and shifting over time.
> "is there any such thing as a government that does not force people to do things they wouldn't voluntarily do?" - no there isn't. That's why it should be limited to protecting other people's rights.
But a lot of the things government does that don't fall within that narrow limit really, really appear to make my life and the lives of the people I care about much better, at relatively little cost (for a broad definition of cost). Few or none of the OECD states seem too bad, to put it mildly. None are unqualified disasters, or even close. Meanwhile a "drown it in a bathtub" government coexisting with an advanced economy (correct me if I'm wrong) remains hypothetical—indeed, I think it's fair to call mainstream the view among experts that governments' special ability to overcome coordination problems and take action to nurture markets is vital to an advanced economy—as do the effects of such a system. So it's definitely going to be an uphill battle convincing me that our government "should be limited to protecting other people's rights", without some other state practicing that and full of citizens telling me "no, really, it's pretty great".
Will we ever see this in reality? No. But, the closer humans have gotten in history the better off they've been.