The Medicare system in Australia works in a sort of similar way to how you describe Japan. The government publishes a list of fees for every possible procedure

The Medicare system in Australia works in a sort of similar way to how you describe Japan. The government publishes a list of fees for every possible procedure and treatment. The medical service provider can "bulk bill" where they collect the money directly from the government, and the patient never hands over any money. Alternatively, the provider can charge whatever they want (because they're a noted doctor, or their office is in an expensive location, etc.), and the patient can then claim the set amount back from Medicare. Bulk billing is so much more convenient for everyone that you'll see signs advertising it on the outside of doctors surgeries, thus the cost of care is kept down. On the other hand, the doctors are free to charge what they want. Sounds like a fair way to do it.