How is it barely better than 'just private'? WHO rankings show that not to be true (the US is 36th at $7500 per capita spend, Canada's single-payer system is 30th at $3600 per capita spend, the UK's 'single-payer' system is 18th, Spain's single-payer system is 7th). The reality is that if you're attempting to minimize illness and suffering, doing that at a larger scale is always strictly better than restricting it to those who are lucky enough to be able to pay. You can of course optimize further but single payer is strictly better than private cover. Two-tier is another valid solution.
I don't know about Canada's and Spain's system, but UK's (where I live) definitely isn't "single-payer" - there does exist private insurance, it allows you to "skip the line", get better diagnosis and probably also better treatment.
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.
