TeenTimesTwo In answer to your original OP:
I think Whitechapelfatberg nailed it in her first response to your question. Classes are large and funding less plentiful than in many/most private schools. Behaviour is mixed. A lot of people go private to avoid that.
Of my dd's state primary class, about 50% went private for secondary. This was despite the state options being pretty good, including a range of high-performing comprehensives and grammars.
Why? Well, some of the children gained entry into some of the independents that are amongst the best in the country, some of the children gained entry into independents with large bursaries or smaller scholarships, some of them chose the independent route quite probably because they (the parents) had been educated privately and they just couldn't imagine doing anything different.
Some who chose the independent route did so because they weren't convinced that their child would cope with the culture of a large state secondary (where they are one amongst many children; where they might not be in the top sets but in ones where behaviour is more challenging). I'm pretty sure some chose the independent sector because their children had a SEND and they found more nurturing, SEND focused schools in the private sector. I suspect one or two chose the independent route because they were a bit thrown by the class mix of the state schools - though I never found a single parent actually saying that. 
As for what the schools are like: my two were in the top streams of state schools. Their classes were, indeed, large. The facilities in dd's school are pretty ropey. The 'treats' (trips, enrichment opportunities) are few and far between. The subjects on offer to study are limited. I write this in comparison with what is on offer at local grammar schools and the independents her friends attend - mind you, her friends are at top independents, rather than ordinary independents.
The school is very mixed, however - both in terms of class and racial background. I wonder if it would be that mixed outside of London? There are lots of very ambitious children in her class, with very ambitious parents. There are lots of high-achievers. The social and economic background of her classmates is very mixed, too. Though I don't think there are any Russian oligarchs in the class - those really do tend to stay in the private sector.
Teaching is rigorous, with high expectations. While I said that the range of subjects offered is perhaps more limited that in some schools, it also has some surprising areas of excellence, which go beyond anything you might reasonably expect. Music is amazing, for example.
In short, if you had the offer of a place at CLGS, and could take that up, you'd choose that over my dd's school. But my dd's school is pretty great, imo.