The difference between the US and the single-payers is that more care is given and better care is given vs. the rest of the world where gov't budgets are constrained so care is constrained.
I'm in Ontario and I can't even get a general practitioner. This is worse than communism.
As for your problem getting a GP, that sucks for you, but that is a local political problem and not a systemic problem with single payer. E.g. in Norway everyone has a legal right to a named GP, and gets one. I live in the UK now, and while there is no guarantee here I've never had a problem finding a GP.
I can afford private insurance here if I'd like it, but I've never had any issues with the NHS that'd justify it. On the contrary the treatment has always been stellar (I've not had to use it all that much myself apart from a "mystery infection" t
If you have the money, nothing stops you from getting supplemental private insurance in Canada either, so I wonder what you mean when you say you can't find a GP. Do you mean that you're unwilling to go private? Or that you actually can't find one either way?