Cleo, well where to start with your sweeping generalisations.. Okay, so all we both have to go on is our own anecdotal experiences, but in your own admission you are only driving past state secondaries, whilst my kid actually goes to one. He also went to a state primary. I'm guessing yours did not.
At his primary school there was a huge ethnic and social mix of families and in contrast to what you describe, we did all mix in the playground. It took a while mainly due to language barriers, but over 7 years of course you get to know other parents.
You might be amazed too to hear it, but my son just like your kids, loves going to school. This isn't an 'outstanding' school, it's classified as 'good', yet he definitely isn't bored and just like your children he has a wide range of clubs and activities to go to. Also, just like your DD, he chooses to get to school at 8am, because he has things he wants to do before school.
Most of the kids I know go to state schools and they all seem v.happy. Fact! My niece and nephew go to private schools and they and all of their friends seem v. happy too. Fact! It's great to see happy kids and in both sectors I'm sure we could find examples of kids who aren't happy too. This isn't about that though, as my original point was about the wider mix in a state school and how important that is.
You seem to be arguing that it's irrelevant, as kids will always stick to others from similar backgrounds. Well I'm sure you're right in some cases (even if just driving past does seem rather random proof!), but even if that were always the case, in most state schools your child would still be in classes with all manner of children, from a wide range of backgrounds. By putting your child into a private school, there's no chance of that.
My son's friends are from very diverse backgrounds and the friendships seem to be based on similar interests, humour and intellect. If he was in a private school he'd make friendships in a similar way. The point though is that his choice now is from a wider range of children. He's learning more about the whole of society at his school by encountering that wide range. I can't see how you can develop that if you're filtered off into the more narrow mix of a private school.