The idea that healthcare can be centrally managed comes from people assuming it's way simpler than it is.
Those are in Canadian dollars by the way; yes, a specialist consult costs the province $100 (and you, $0).
It's hard to hear that you're trying to pre-solve cost problems you don't know you'll have if you were to switch to central management when the US is currently the most expensive healthcare system per capita by a lot (like, double), and is ranked 36th by the WHO. Far less expensive systems per capita are ranked dramatically higher even though they have a fixed fee schedule. Sure, there's room to improve, but maybe knock out the low-hanging fruit first.
Further, if that's too distasteful, there's other ways to make this work; the Swiss system forbids private insurers from making profit on the basic administration of healthcare, and caps individuals out of pocket expenses. I'm sure the insurers would find a way under such constraints.