You're entertainingly deluded if you think Canada is a single-payer system.
We have a tiered healthcare system, it's just that most people don't seem to acknowledge it.
I have partial coverage from my province. I have partial coverage from my employer. And I cover the rest out of pocket.
My mother-in-law, having turned 65, but is still working, has even more payers: the province, her employer's health care coverage, the provincially-mandated health care coverage (different pocket, not sure if she pays for it or not), and then the rest out of pocket.
There are health-related items that are fully covered by provincial plans, some that are partially covered, and many that are not covered at all. For the last two categories, private health insurance can cover all, some, or none of the extra costs. If you have multiple health insurance providers (e.g., two different employers for a household, usually they cover the employee and their spouse and kids, so you have two insurances covering the household), there is some sort of duking it out for who covers what, but, in the end, you usually end up with the higher percentage of the choices being covered somehow. And then, whatever is left, is your responsibility.
I go to the doctor with a cold. The province pays the doctor for my visit. He wrongly prescribes me some antibiotics. I go to the pharmacy, get the pills. The province doesn't pay for any of that (though they play a role in regulating the drug costs). My employer's health insurance pays some of the drug cost (the percentage widely varies on which drug it is) and none of the pharmacy fees (other insurances do pay for pharmacy fees). I then pay the rest, never less than 10% due to the plan I have with work.
If I then spend more than a certain percentage of my pre-tax income on health expenses, I can submit them against my taxes for a further refund, though I've never hit that amount, personally. I'm sure lower incomes could easily hit that.
If I go to the optometrist's, the province pays nothing for my visit, but all of my children's visits as they're all under 18 years of age. I submit to my employer's health insurance for my visit and any and all prescription eye wear that results, including for my children.
If you cannot get health insurance from your employer, or you cannot afford to get insurance on your own (here in Alberta, there is a cheap provincial-run insurance available for purchase, not sure about other provinces, and no idea what kind of coverage it gets you), you get to pay full costs for chiropractic and vision care. Dental visits are also not covered, or any orthodontic care. If you're poor, you're going to be stuck with bad eyes and bad teeth. Even in Canada. Because you're in the bottom tier of health care. Which itself is because we have tiered healthcare.
Sure, emergency access is paid for. But same in the US - effectively. If you can't afford it in the US, the hospitals eventually absorb the cost, by law. In Canada, the government absorbs it. However, if you can't produce your healthcare card, you're still responsible to pay for it - tourists and out-of-province patients don't get free rides. (However, when I was in Toronto a couple months ago, had I required health care during that time, my home province would have covered the costs same as if I were at home. Which, again, means not everything is covered.)