The US military provides free care of this sort to members and their dependents. They get paid a salary by the government, but don't charge the patients for care per se.
I'm quite confident that's part of why therapy did me so much good. He wasn't going to make less money by actually being effective.
Well, there's a lot of situations in the US where the need for the doctor to be paid becomes a blockade to getting care, particularly for those in need of mental health.