Incorrect about Canada. Doctors are prohibited from practicing in both the public and private hospitals. There is almost no private surgery - people just go to the US.
That's just stupid. It just makes sense to have a fast lane for paying patients because that shortens the queues to the public surgeries as well.
That works pretty well in Finland. One of my parents needed a cataract surgery this year, all cased and diagnosed by doctors in public healthcare. The actual surgery would've been maybe six months from now. Private clinic -- three weeks. My parents didn't even go shopping for another clinic, waiting three weeks was no problem. It also cost much less than they anticipated, a quick routine operation not worth the wait.
