To be honest I don't see the value of private schools
What about religious schools? They can't be public by law.
I don't really see the point of religious schools either. schooling isn't about religion, it's about learning. there's a wide range of outlets for religion, the outlets for schooling are probably considerably more narrow.
To this, along with your other comment about non-academic learning being for outside of school, I'd just like to re-iterate that schooling isn't just about learning science or English - it's about learning for life, something which religion (for some anyway, I do profess a fear for hardcore fundamentalist religious schools) often provides for. I can vouchsafe for this personally - the religion classes at Melbourne Grammar made a massive difference to my life (and not in the sense that I became religious, because I'm not).
To your points about overgeneralising being bad, I agree - nevertheless, you're restricting all of your arguments to a small sample space. Although you may know a lot of people who want to do well, and although your friend's schoolmates might be laidback and apathetic, from a logical perspective there's no reason why there wouldn't be a larger percentage of "motivated" kids at private schools. Again though, overgeneralising is bad, and I'm not saying you won't get these kinds of kids at public schools too (look at some of the people on VN, for instance...), but the atmosphere in private schools will often just foster that kind of attitude more easily.
I think Russ and Appianway have put forward the best views here. In the grander scheme of things, it's not like private schools are "lolzt3hawesomz" in comparison to everything else by a long shot; nevertheless, it's undeniable that they're at an advantage, and it does come down to whether or not the cost is worth it for attending.
On one last note, Appianway does also bring up a good point - although it's arguable that it's necessarily a good thing, private schools tend to attract a lot more internationally/socially focused individuals, and in an environment where people are casually applying for Ivy Leagues/Oxbridge, you're bound to set your sights a bit higher...