...snip Libertopian theory...
What all of your theoretical musings ignore is reality. Your position is based on your ideology, not on the real world.
In the real world a private healthcare system inevitably ends up with poor people not getting healthcare. Even if we accept the fantasy about charity covering some of this there can be no doubt that some people would still suffer from lack of healthcare. This is not a world we should choose to live in.
In every other modern wealthy country there is some kind of safety net and in many countries there is Universal Healthcare, all provided through taxation. It turns out that is much much much cheaper and efficient to provide UHC through taxation. In the UK we pay less per capita for UHC than you pay in your taxes for just medicaid and medicare.
Aside from this the benefits to society of UHC are huge. You can remove a huge amount of stress and uncertainty and suffering in society by having UHC. It's like education, there are knock on good effects from having universal coverage. Just like we don't accept sending children down mines anymore we shouldn't accept letting poor people die and suffer for lack of healthcare, its bad for all of us. Plus you can have this cheaper than you pay in taxes already!
Of course under UHC wait times for major operations will increase relative to being well insured right now. (Contrary to your assertions the quality of treatment under UHC is arguably better as it is driven more by patient needs instead of profit needs.) So there is still a place for top up insurance in a UHC system. In the UK more well off (or insured through work) people can get private healthcare. In practice all this does is allow you to jump a queue and maybe get a private room, the actual operations will be done by the same surgeon as you would have got on the NHS. Emergency care is always provided by the NHS as they have the best facilities and doctors.
Its very telling that there are almost no examples to be found of private healthcare users in the UK getting treatment privately that they couldn't get through the NHS; the only examples there are seem to be of extremely experimental treatments that there is a lack of definitive evidence as to efficiacy. If there really were death panels in a UHC system we should be able to see examples of people getting cancer treatments privately that they couldn't get on the NHS... where are these examples?