It sounds like you’re describing a country with a public-private hybrid system. Certainly in the UK and the Nordics, the systems I’m most familiar with, there is no “getting back as much as possible from the insurance providers”, as there are no insurance providers. The hospitals are run by the government, funded by regional governments, and have fixed yearly budgets, not treatment dependent billing. The only exception is some privately run GP clinics that may bill per service rendered.