Are those market price caps, or are those caps on what the 'single payer' in a 'national healthcare system[s]' is willing to pay? I was under the impression that in these systems, the 'shortage' was created by the fact the single payer was not willing to pay the free market rate that would clear for these services, therefore there is an undersupply of services provided to the 'single payer', not that there was usually an actual market cap.

Typically in these countries you actually can get health care as long as you pay privately yourself and don't go through the 'single payer.' A price cap would mean that no matter how much you're willing to pay, you can't pay over the cap, which is much rarer than the presence of 'national healthcare systems' that merely won't pay over the supposed soft 'cap'.