Though something that gets lost in the "aargh, socialist death panels" style discussion in the US, is that there are plenty of market-based mechanisms within the most "socialist" health care systems around the world, and there's a whole bunch of weird, anti-market bureaucracy in the American system.

So you're right that talking about rights is unhelpful, I'd go further and suggest that just making healthcare provision better, rather than more/less market-based should be the main discussion to further avoid the conversation bogging down into "reasoning about ideology, rather than outcomes".