I don't think this has much to do with being a democracy. I, for one, wouldn't trust our federal government to competently run an efficient, most uncorrupted healthcare system for all.
The incentive structures that have built up around US politicians simply doesn't leave any room for it to realistically happen. Until the incentives are changed I'd vote against nearly any major government program.
That would entirely depend on the coverage I could get and how it compared to private. If Medicare is changed over the next few decades to look more like Medicare Advantage, aka Plan C, you couldn't pay me to deal with that.
Edit: its worth noting that your question for whether I'd take medicare is a separate issue from my original point. If the existence of Medicare as it is today was on the ballot, I would vote to get rid or drastically change it. If the program exists regardless of my opinions of it, my choice to take benefits from it is entirely a question of means and comparative benefits of all the options.
The incentive structures that have built up around US politicians simply doesn't leave any room for it to realistically happen. Until the incentives are changed I'd vote against nearly any major government program.