I don't think you should downplay the scope of "not publicly insured" services in Canada, especially after the last couple decades of aggressive privatization of once-insured services.
It varies from province to province in its specifics, but the majority of working-age Canadians have private insurance through their employer. This is expected to cover everything from physiotherapy to psychotherapy to prescription medications.
People without such insurance have a limited patchwork of public programs and usually either have to pay themselves or go without.
Case in point: A friend injured his back, and while the surgery and doctor's consultations were covered, everything else including medication and physiotherapy was not. He ended up spending tens of thousands out of pocket to be able to stand again.
The private health insurance market in Canada is closer to a truly free market and isn't regulated like in the USA, and that means the insurers happily and routinely deny coverage for pre-existing or self-inflicted conditions (as the insurance contract may define them).
Another case in point: A friend of mine is HIV+, and was denied group coverage through his employer. He pays thousands of dollars out of pocket for antivirals every year.
When the public insurer doesn't cover you, it's an absolute wild west of unregulated private insurance coverage and usually boils down to pay or suffer and die.