Short answer: no.

Healthcare providers are not allowed to charge publicly-insured patients for covered services. Insurance companies are not allowed to insure for covered services. So for covered services (e.g. lab tests and doctor visits) the only available options is public. For non-covered services (e.g. ambulance, dental, prescription drugs, etc.) you pay out of pocket or use private insurance.

Many healthcare providers themselves are private. For example, doctors are independent contractors, not government employees, and their offices are private businesses. But they get paid for services by the government (why we call if a single-payer system rather than a public system).

Maybe things have a changed in the past few years, but I knew an orthopedic surgeon with a private practice in Vancouver. The workaround was that he charged a flat hourly rate, not a per-service fee. He specialized in knee surgery and also worked at a local hospital, so you could either wait 3 to 6 months for him to fix your knee at the hospital, or you could book him sooner at his private practice. That sounded like a win-win to me.

A. It's not a true win-win. He is not creating and selling extra treatment capacity. The time he spends in his private practice is the time he would have otherwise spent in the hospital. The same number of patients end up receiving the same number of services. But now, instead of need and urgency determining who gets the services first, it's money.

B. It is not a loophole. The federal government fines provinces who don't rein in private billing. British Columbia loses millions of dollars in Canada Health Transfer every year because of lax enforcement though. BC lost $15.9 million in 2018 and was on track to lose $32 million in 2019, but they started enforcing the law better.

https://lop.parl.ca/sites/PublicWebsite/default/en_CA/Resear...?

https://www.cbc.ca/news/canada/british-columbia/extra-billin...