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.