Agree op to a degree.
I've found our state cohort very narrow really. I had a mix of my own schooling and found it all good and bad for different reasons.
I was put into a private for a few years and classmates came from all over the area, near towns, countryside, so going to their houses was fun. Also many pupils came for a short or longer time due to dp work so we had a girl whose dad worked in oil Nairobi, another from the States, Saudi, etc and I loved it when there was a presentation about where they were from.
In terms of intelligence diversity, it wasn't selective at all so we had students that struggled, and those that didn't.
4 girls in my small class had some sort of sen. One I remember definitely dyslexia but not sure about the others.
In terms of wealth, a few had 6 bedroom homes. But nothing like helicopters 😂.
Also many were there funded by different means, at least one scholarship, divorce, lived with mum in very teeny house, dad paid school, same with other pupils, whose maybe gp paid the fees but their own dp were low earners in modest homes.
Also and I'm not sure how to phrase this... All sorts of people make money... People who have little experience of private schools do seem think they are full of hooray Henry's with pink corded dad's and rings on their pinkies...
In eton, perhaps but my school we had a wealth of characters who could pay for their dc to attend. One rumour was dc from crime family! I guess not everyone there (including me) was posh posh?
I remember one poor girl in year above, her mum was h addict, she had a nightmare time of it. Dad not around but v wealthy, in music...
So yes, I do think there is or can be much more diversity in not too expensive, local private schools.
I think once you get to the enormous sums that eton require, whilst I imagine they do get cultural diversity... Probably not so much access to economic diversity.