While some of that might be true for some countries with publicly funded dental care, it’s certainly not true for regular healthcare. Doctors are paid fixed salaries, there’s no incentive for them to perform unnecessary procedures. Quite the opposite in fact, departments have strict budgets and so they have a strong pressure to ensure all procedures are medically justifiable, and cheaper / simpler / preventative options have been tried first.
Individual doctors working for a clinic are paid a flat salary, true - but the clinics themselves, or the small doctors running their own office are absolutely dependent on accurate billing to the insurances. They have an incentive to optimise their margins, even if patients are not paying directly, and that means both trying to spend as little on care as possible while getting back as much as possible from the insurance providers.
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.
This is true, but it’s also assuming the robot has to be human shaped. A robot with 4 extendable arms and a gyroscopically balanced cabinet in its chest wouldn’t have too much trouble with that task.
How many other tasks is it not suited for now though? If we wanted a device to automate limited specific tasks then we don't need advanced highly advanced robotics. A human has strength, dexterity, high balance, very sensitive tactile touch, can move both fast and slow, is very compact, etc. A robot has to have serious tradeoffs just to hit two of those things. A robot arm that is capable of threading a needle is likely not suited to chopping some wood or carrying groceries. And if you just add a bunch of different arms for different tasks it becomes way more complex, expensive, heavy, and inefficient.
Source: Also a European, spouse is a doctor