I live in Australia where there is a Public Health system funded by Medicare, plus a Private Health system funded by Health Insurance. There is a 1% Medicare levy for those with Private Health Insurance, 2% for those without: both levies are paid as income tax, so the more you earn the more you pay.
When you need a surgical procedure, you must choose whether to go Public or Private: Private fully covers the cost of the hospital room (around $1000 per night) but only part-covers the Surgeon and Anaesthetist: there is always a gap payment and somethines that can be large: $4000 or so.
Public is free, but there is a waiting list for most procedures of up to 6 months.
As you say, you can have any two of Quick, Cheap and Good Healthcare.
Private is quick and good, but not cheap.
Public is good and cheap, but not quick.
I disagree with you on cost benefits of Public Health systems though, and stand by my statement that “Public Health Hospitals compete with private ones and drive prices down in countries with Public Health systems.”
The USA pays twice as much per person per annum on Healthcare as the rest of the Western World on average for a comparatively poor result: the life expectancy for US citizens is 43rd in the world and is the only Western Country were the life expectancy is falling. Something is very wrong in American Healthcare.
Perhaps a balance of Private and Public Healthcare is worth considering.