Fwiw, a lot of European healthcare has both a public and private option. You may pick the private options because they are "better" in some ways (e.g. more modern clinics, shorter waiting times, or sometimes just better care) which still leaves some wealth gap, but usually means no one goes bankrupt to cure cancer.
Usually the medical part of it all is strictly worse in the private sector (at least in my country) because the public system has a pretty strict competitive exam to get in, whereas profit driven private companies hire the cheapest doctor they can get.
Not everybody realizes that and they often fall for the single room in the hospital.
Shorter waiting times is definitely a thing though, especially for non life threatening conditions.
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.
In the UK a large proportion of the doctors are the same. Sometimes even using NHS operating theatres, or with NHS trusts running the clinics, as they are allowed to run for profit services to supplement their budgets...
