Heck, here's a news article about the Quebec government outsourcing some work to private clinics to clear COVID-related backlogs. For there to be $100M worth of contracts, there must be a decent number of clinics.
Doctors are not able to work in both the public and private systems, they have to choose.
There are a few clinics that provide care entirely outside the public system but since the demand is low they are rare beyond MRI clinics in a few provinces.
But please, send me links to the dozen of private clinics (dont work in the public system) that are 30 mins of where you live.