> (like eg the British NHS does, modulo the option to go to a private hospital for enough money)

IME it's more complicated than that.

Firstly, the NHS allows consultants to do private work, making use of NHS resources, including beds, radiography and theatres. Secondly, a consultant can refer you back to the NHS for treatment, effectively "jumping the queue".

I would never rely on a private hospital. They never have an emergency department, and emergencies happen in hospitals too. Many private hospitals can only treat certain types of case; I wouldn't want to be stuck in a hospital that couldn't treat anything that happened to me.