One of the things that we should keep in mind is that the Canadian public education (in general, though it is a Provincial jurisdiction) system tends to provide a more qualitatively uniform experience across a jurisdiction than south of the border.I have two cousins, both over 30, now, whose father is a doctor. They both went to private schools since day one. The parents didn't provide much guidance. I guess they thought that the private school would handle most of that. One ended up failing to get into medical school, disappointing the father (it's not easy to get into medical school, even if your dad's rich and you go to private school), and ended up going back to school to do law, after wasting some time, so a partial success. The other cousin, fucked around a lot, went to community college, and ended up selling cars. A huge failure in daddy's eyes.
My son is 13, goes to public school in a good area. I used to help him with his homework since I'm good at math and science. Now, he doesn't ask me for help much. The little fucker is going to be smart than me!
The biggest X factor is the actual kids attending a given school. A school in a bad area will have kids that have difficult social and economic circumstances, but the school itself generally won't be severely under-resourced to boot as is often the case in the US. I is generally possible to get a decent education at pretty much any Public School.
Can private do better? Connections wise, probably. But second generation nouveau riche often go off the rails so I'd suggest your kids (or you) get to know their parents instead... If the opportunity is there, and the school has a good rep, why not? Though it's far from necessary in the general case. Public / Separate / Private all produce decent outcomes, with 2 of the 3 being 'free'.
Involved parenting will pretty much always trump any schooling in producing positive outcomes for children...