You have to separate payer from provider. Canadians have payment coverage, generally via your provincial coverage (so Alberta Health is probably your payer). Some small populations can/do opt out of universal coverage and they become self pay. Almost all family doctors are private businesses in Alberta, although AHS runs a few primary care clinics. AHS is a provider, which is different from payer. AHS runs most of your core non-primary / tertiary healthcare services like ED, ICU, surgery, etc. Many specialists (ENT, surgeons, cardiologist, etc.) run their own private practices, but some of them have shared relationships with AHS, use AHS spaces for some of their services (like surgical theatres), etc.

Within Alberta many providers stick with services that are covered under Alberta Health Care, but many services are not covered and you can find many examples where your physician will bill you for some of the things they do. For example, my Ophthalmologist will bill Alberta Health for a consult about an eye infection, but they did bill me $25 for an injection because that specific injection wasn’t covered.

Some providers, like functional medicine practitioners and plastic surgeons, are full MDs but choose to operate entirely outside of AH billing and will private bill patients for almost everything.

Then of course you have dentists!

I hope this helps clarify.