My post-Soviet homeland has that evil socialist public healthcare - the remnant of the Soviet system. At the same time, we also have a parallel layer of healthcare - private clinics and hospitals. State-run healthcare is free, private is obviously expensive af. Private clinics are generally for the rich - most people only go there if they need to get a check-up/get tested right now without delay, though almost everyone prefers private dentists over state polyclinics. However, if you need to pass medical check-ups for things like applying to a job or getting a driving licence, you're required to go to state-run healthcare service, as they are the only one authorized to do that under our laws. Hardly anyone has health insurance, because hardly anyone has any insurance in general. Some private businesses like big foreign-owned factories has their own doctors - service is obviously limited to employees.
Public healthcare is very much plagued by queues - you can generally get an appointment only at very inconvenient times. If you're working full time, you'll have to take a leave just to go to the clinic. Most people in queues are either pensioners or people applying for jobs/driving licences. Doctors are often overworked and their pay is not as good as it should be. The level of equipment depends on the place. Some recently renovated polyclinics are very nice and have things like appointments via Internet and electronic queue systems, like a bank. Some look like they haven't been touched by renovation since the 1970s. It depends on how lucky you are - you're assigned to a particular public polyclinic according to your place of residence, and then to a particular GP according to the street you live at. A person living in the town of Examplegrad in Leninsky District at Red October Street goes to a check-up in polyclinic #1, to area 15's GP doctor I.I. Gulagovich, who then appoints treatments and sends them to specialized doctors, if necessary.
Private clinics look more like their Western counterparts. They adhere to the same regulations by the Ministry of Healthcare, but since they are for-profit institutions, they can spend that extra money on top of the line medical equipment and better pay for doctors. Thus, the quality of service is obviously better, but even a simple check-up is costly. On the other hand, no queues - the service is too expensive for most people.
But there's a very big catch. Even if you go to a state-run polyclinics, the attendance is free and the therapy is free... but the prescribed medications you get at the pharmacy are not! They weren't free even in the Soviet Union, except when you were hospitalized (not sure if that still applies today). Some basic medications like aspirin, activated charcoal, certain flu and heart medications and some antibiotics are manufactured locally and are heavily subsidized, but the absolute majority of medications are imported. They cost an arm and a leg, because you get Big Pharma's greed + customs dues at the Schengen Area border + the markup of pharmacies themselves (most of them are private-run). Vaccines are imported - either from the EU or Russia. If you are an old person with hypertension, for example, you can't live without a device for measuring blood pressure... which are also imported from abroad! Thus, this paragon of socialism is not as free-of-charge as you might assume.
Oh, and neither public nor private hospitals can cure many extremely rare illnesses, particularly nasty genetic disorders. If your child has one, heaven help you. Hospitals in Germany or Israel demand astronomical amounts of money for treating such stuff - as a result we have a lot of charity drive ads on TV like "Insertnamehere was diagnosed with insertdiseasehere at age 3. Doctors at home were helpless. Doctors in Germany can treat them, but the cost is 500 000 EUR - more than the parents can afford. Please help them save Insertnamehere by sending an SMS to the following number..."
What does this have to do with America?
I don't know why you can't have both public healthcare (federal-run, state-run, whatever) and private healthcare systems at the same time. It'll still fulfil important American traditions like screwing over poor people, benefiting the rich and lining up the pockets of (pharmaceutical) corporations without any disruption to everyday life or political system. With the amount of money in US budget, I'm sure America can create a public healthcare service no worse than ours even without health insurance. And I seriously doubt the federal government would railroad people to visit public clinics like our government does, which means you get to choose where to go. Why haven't America just cut the knot already, I have no idea.