I think I would be fine with healthcare paid by Government - but not actually run by Government. They suck at it.
That's basically what Obama-care plans are, which in my clinical experience are.....not great. The government doesn't administrate those plans, but they still run them in the sense that they design the healthcare policy. Instead of "not run by the government" do you perhaps mean "the government doesn't direct policy for the plans?" That would put the government pretty well hands-off, except why would the government foot the bill for something it has no say over? If the government pays for a plan but other entities design and administrate it, that's basically subsidizing private healthcare. Which I'm also generally not in favor of.
I know it's tempting to see ways and workarounds by which the government can make healthcare better. In reality, I don't see a good avenue for it, and I'm not impressed with the outcomes with attempts, anywhere--not just here--for government to make it "better," neither for the patients nor for the providers.
Medicare and Medicaid, I admit, are not bad (imo) for what they are. And at least at the moment, they're necessary, without better widespread alternatives, so I don't advocate for ending them. For various reasons, they won't scale to the entire insurance sector. As is, they provide a reasonable, mostly-workable option for both patients and providers. Medicaid, less so, but it's better than nothing.