That's really interesting AliceAnnB, and I would probably agree with you on a lot of the issues you have raised; about Fortismere and our education system anyway.
I don't know much about APS, but it sounds very much like the London Comp I went to. You probably didn't see the children who would argue back at the girls on her bus, or the ones that don't fight.
Anyway, what you are probably finding is that most parents don't go private at secondary in muswell hill for a variety of reasons. They pay tutors to get their children into grammar, or they are happy with Fortismere. Maybe state primary schools in areas where there isn't such a highly regarded state secondary have a higher level of children going private for secondary. MN164 seems to have covered what you need to do, and I imagine there are tutors who specialise at getting children into particular schools.
If it helps, I think there is evidence that class size is not actually an issue in educational achievement. In other words people believe it to be an issue, and think it makes sense that it would be, but actually the scientific evidence doesn't show that it is. Not much help without the link though, which I may remember and post ;)
Could you go private all the way through to alleviate your concerns?