I know a good deal about the Middle Ages, being part of the SCA. And indeed, it did work pretty well.
The problem is now is that it's not just 1 or 2 people involved with health of the area. It's the doctor, the 2 nurses, the 2 receptionists, the x-ray techs, the lab techs, the pharmacist, the pharmacist techs; all for a call to the doctor.
In hospitals, it's so crazy how billing works, that most hospitals do not actually know how much X costs. instead, they just bill exorbitantly to anyone without insurance, and insurance gives them the agreed upon price.
Drug companies are also really messed up: the US ends up subsidizing the rest of the world's drugs, all the while biasing on maintenance based drugs over cures.
The FDA also has its fair share to blame. They control which universities are accredited, and directly control the flow of doctors purely on the say-so from the AMA, a doctor union.
Considering the complicated nature, insurance covering the nation, aka socialized care, makes the most sense. Tearing down systems like I said above really isn't sustainable.