You are wrong. The single payer in Canada is each province's respective health insurance plan. Anything listed on their "schedule of benefits" can only be paid for by the province.
Providers are actually in many cases like contractors... most family doctors, for example, bill for services rendered and are otherwise free agents (unlike in the UK).
Correct me if I'm wrong, but doesn't Canada have publicly funded providers? I'm all for single payer, but with a completely privatized provider network. Make the free market compete for the government's dollars. Technology should theoretically reduce costs because it more automation generally means higher profits.
provider != payer