It's quite country dependent. For example in Hungary my understanding is that many doctors have both private and public practices, but private clinics often don't have expensive machines. Doctors in public hospitals are severely underpaid so they have strong incentives to move you to private practice.
In Italy doctors also have public and private roles but can practice privately in public hospitals, which is weird but was an attempt to avoid losing them to private clinics, for the same reason. You also have private clinics administering public healthcare with a minor markup paid by the patient, and the base rate paid by the state, which isn't a thing in Hungary for example.
It is in general for non-life threatening conditions that there's such competition tho, I agree.
