There's a limited amount of health care to give because there is a limited amount of people willing to provide it. There is no concept of "right to health care" that doesn't require slavery.
You can argue for universal health care but arguing that people have "a right to health care" makes no sense.
Imagine there is 1 doctor. Doctor doesn't want to treat patient (could be sick, could be tired) but patient "has a right to health care". The only way to honor that right is to force the doctor to treat them.
I'm not sure how "every government worker is a slave" fits. The things most government workers do have nothing to do with "rights". As an example there is no "right to protection" even though we have police. There are rights that prevent the police from doing bad things but there is no right that forces police to help anyone.
What said was claiming people "have a right to healthcare" requires slavery.
Universal Health Care != "right to healthcare"
I can't imagine that the homeless are adequately covered. I think it's painfully, ridiculously absent as well.
To be clear, I'm not in that camp, but there are rational lines of thought to oppose a state-run healthcare system.
(And also to be clear, I can't personally see how it could possibly be worse than our current system)
(I personally believe we should have state-run healthcare, but universal healthcare != state-run healthcare.)
Sure, the day-to-day administration of such a system could be in a single agency, or distributed among private providers, or some blend. But the core thing that opponents fight is still there.
Success would be wholly dependent on the implementation in any case. Our current implementation is a local minimum of effectiveness, maximum of overhead. There are better solutions with either public or private bases.