There are private healthcare facilities which are tax funded and there are private healthcare facilities which are not, patients pay out of pocket.
...for non-essential services, which I'm sure is big business in Canada. For example, Lasik surgery was for a long time not covered by insurance and deemed "non-essential" or whatever because glasses exist. So Lasik was big business all over the place. Patients paid out of pocket*.
* For all I know this is still the case
But there aren't that many privately funded because there's not a huge demand.
It seems like there are plenty which are privately funded which cover non-essential services (like Lasik I would guess). The Canadian Medicare system says that they need doctors, there's a doctor shortage (if headlines are to be believed, which is probably not safe). They therefore make sure that the most doctors can make for services covered by Medicare is through Medicare. They make it such that there is no other option really but to use the system. Technically, maybe there is a way, but they make it so painful that nobody will do it.
As a result, you can't get out of line. And that was the idea to begin with.
Edit:
To sum up. It is illegal in almost all of Canada to accept private health insurance. And while private providers exist in Canada, they are regulated under medicare (including the rate charged) if they offer services offered by medicare. Bypassing those regulations results in state fines for the provider.
You can choose your "doctor" but only to the extent that you must choose a PCP doctor if you're getting essential services offered by medicare. And if you're getting an essential service, you're getting it through medicare (for much of Canada) because medicare has squashed the alternatives (in much of Canada). You cannot choose to see a specialist through the medicare program, you must first convince your PCP that you need one, and their function is to be the gatekeeper.
You can technically go to a specialist directly, but they'll get docked pay if you do, so they won't take you. You can't find a provider for essential services that is entirely private (and by this I mean is bypassing the wait lists) because they're banned from accepting insurance and regulated under medicare anyway (in most of Canada). And that's exactly what people want to stop, folks from buying their way around the wait list.
So... for much of Canada... it is as I said. If you're transgender (aspiring?) and want a gender change, you need to convince your PCP that it's medically necessary. And if you can't, you're shopping out of country (or at least, somewhat similarly, out of province). If you can, get in line.
Edit:
I mentioned this issue to my wife and she brought up an excellent point (this was yesterday). Breast reduction surgery could be an essential service. But it could also be entirely cosmetic. Does this mean that if one doesn't
need breast reduction surgery in Canada that they cannot get it? Because it's considered "covered" but also not prescribed for that patient? Surely there's a way around the cosmetic surgery that's sometimes-required-but-sometimes-not problem right?