I grew up in Central London (SW1). Apart from access to museums, history, theatres, concert halls, sports facilities and so on, the extensive public transport system meant I was pretty independent once I got to secondary school. (In fact, we were allowed to travel independently on the tube once we were 8, but this was the 1980s. Not sure if even my super-liberal parents would allow that now). Now (rural W Yorks) I just see years of driving my dc everywhere ahead of me. Not because they are special, but because buses are minimal here!
My primary school faced the Natural History Museum and my secondary school was 10 mins walk from the original Tate Gallery. Both schools used these to support work we were doing.
And big schools may put people off, but if you grow up in a capital city, large groups of people is what you know. Whilst my academic achievements at school were mediocre at best (I improved dramatically at university), the social and cultural mix of children I went to school was was amazing. My secondary school had pupils with 46 first languages between them; with pupils including a hereditary heir and some from the biggest sink estate in Europe at the time, and every socio-economic increment in between. I probably have the most cultural and racial understanding and compassion of anyone I know, because it's what was my "normal" from the beginning. I knew sane sex couples as friends of my parents, went to tea and celebrations with Jewish, Muslim and Sikh friends. My best friend was African. I'm not saying that multiculturalism is unique to London, but London is a huge and diverse city, which exposes its residents to experienced my own dC just simple do not come across in their day to day life. (I am also not saying that this is a good thing, and it is something we are working on with our country mixe).
And the point about green space/wildlife has done validity. I live in rural W Yorks now, in the middle mixed farmland. All we can see for miles is green rolling hills and woodland, but we can access very little of it! Nothing like the hundreds of acres of Richmond Park I grew up with, Battersea Park that I cycled through to get to school (moved house when I was 16), Streatham Common where we rode, Hampstead Ponds where we swam. And there was plenty of wildlife - rabbits, deer, badgers, foxes - in all of them.
It could be that they are Yorkshire folk, but my dc are never happier than when they have a plate of meat and 2 veg with gravy. No wandering Brixton market for hot patties and plantains for them!
So, house prices aside, I've never quite understood this horror of raising children in London.