The "just public" system doesn't work for similar reasons as communism doesn't work. (1) Lack of incentives and (2) corruption. A paid-for healthcare encourages innovation (you can sell expensive new treatment to rich people who want to live longer, when the technology is mature it is then distributed to the masses), and prevents corruption - e.g. in Slovenia, which is "almost public" (i.e. the private healthcare system is severely limited by the government), there is rampant corruption - if rich/powerful people can't "jump the lines" by paying more, they will (try to) jump the lines by utilizing other forms of power (threats, favors, gifts, ...).
It's easy to shit on US/Switzerland, but keep in mind that their "expensive" healthcare systems also fund a huge amount of farmacy research that then "trickles down" to other countries.
Hint: communism doesn't work.
Also, the reality is you have to prioritize. A public health-care system has a limited amount of money (by definition) and needs to spend it in a carefully balanced way. E.g. you'd rather operate/save a kid than a 90 year old person. Another example is, governments "advise" citizens to do preventative exams based on cost-efficiency, not on "best quality healthcare" - it's simply unrealistic to perform a mammography on 50% of the older than 25 population every year. But if someone wants to do that, and can pay for the costs themselves, why would I, you or the government limit their freedom?
But also, there's no reason why you can't have private insurers providing better options on top of a public system. In fact, most countries do exactly that.