Earlier quoted context omitted.
The US healthcare system is wildly complicated and inefficient because it is a double-bureaucracy; pubic and private. The government bureaucracy makes a bunch of rules and also provide healthcare through Medicare/Medicaid. The private bureaucracy compete with each other, and hospitals, and pharma companies, ect. Many of the private health providers are for-profit and lobby against rule changes that would reduce compl…
Canada is the only country I know of that has only one system - public. So your claim of public + private being the issue makes no sense when almost every country has that.
For procedures covered by provincial health programs, doctors in private practice (GPs, at least some surgeons) give the service to the patient and then bill the government for that amount. For services that aren't covered by provincial health programs, GPs/surgeons bill the patient.
Provincial health care programs cover anything medically necessary. My general understanding is that you cannot pay to have a service done which is covered by a provincial health program. The intent is to prevent the establishment of a two-tier health care system.
It's probably less complicated than the US system, but it's still complicated.