Issues with private health care:
1) Private health insurers haven't been able to run anything efficiently as long as I've been alive. My premiums go up by double digit percentages every year because "costs" keep increasing, but at the same time, insurers' profits go up by double digit percentages every year. It seems like this is simply a money transfer to health care executives.
2) The more private companies run our lives (especially something like healthcare), the more they will be able to dictate our choices (what we eat, risky behavior etc). Oh wait... corporations already do dictate what we eat, what sort of medicine we use, where we work, how we communicate, how we move around, what we wear...The only difference between corporate control and government control is that we actually have some say in how the government controls us because at the end of the day we control the government.
3) I haven't seen on person that supports privated health care tell me the downside. I know they exist and we need to know the risks and talk about them openly before we stick with such a horrible system.
But anyways, the downsides to public health care: cafeteria health care (i.e., standardized treatments) which is good for most patients but not for the few who have nonstandard variations of ailments; higher taxes; less personalized care as a consequence of standardized treatments; potentially reduced choice of health care provider(s) (but if the entire system is UHC this may not be the case); potentially longer waits for medical treatment that cannot be handled in a day (i.e., anything that cannot be handled outpatient or by nurses or nurse practitioners).
And the downsides to private health care: all of the above, plus: risk of claim denial for valid claims (though admittedly less of a risk now as a consequence of numerous state and federal laws cracking down on claim denials); double-digit increases in premiums every year despite no corresponding increase in costs (this is actually happening, and has been happening, every year for the past decade); reduced choice of health care providers (which is actually the case now, i.e., in-network vs. out-of-network); ruinous medical bills; unnecessary tests and treatments; private health information being shared with "business partners"; inadequate medical facilities as a result of lack of capital investment due to premiums being redirected to marketing or executive retention.
It turns out the downsides of a private health care system outweigh the downsides of a public health care system, since you have all of the same downsides but the additional downsides that come with a profit-seeking business. And as history has amptly demonstrated, cost-cutting is not one of the benefits of a private health care system (costs are cut in a way that benefits the insurer's profits, but in ways which are detrimental to the insured).