OP, it's a shame that you have had a lot of parents provide their assertions without experience.
I, and every other parent of black children that I know, without exception, considers the ethnic makeup of a school in depth.
I know people who are considering offers for some of the counties most academic schools (think KCS, Eton, Winchester, NCLS) and the lack of black (not BME but black) children is a major conflict.
Whilst BME is an inclusive term, it doesn't take into account the different cultures that exist within it. In our school, a large proportion of the black children have faced racist incidents (often naive, media encouraged, parental ignorance-based) and comments. And, because these environments have historically had fewer black people in them, a lot of these stereotypes have gone unchecked.
Black people have a unique and individual set of racial stereotypes made against them, as does every other group, but a lot of the stereotypes are what one might say are bottom of the barrel. Think thieves, promiscuous, more aggressive, less intelligent, less wealthy, etc.
In contrast, although still racist, a lot of the stereotypes that exist around say South Asians, for instance, are well-prepped, very academic, very engaged parents (tiger mum-ish) and very ambitions. Whilst this is true in part, this is also true for other racial groups and not true for many South Asians.
As reluctant as I am to play stereotype Olympics, I know which group I'd rather fall into.
A lot of these schools are underrepresented because these exact stereotypes lead to a whole host of identity issues (as above) and I know in my experience, that being around people that looked like me helped, in part, to deal with the daily microaggressions of society.
Whilst the academics are undoubtedly amazing, what is the point if the children come out with identity issues and a lack of self-confidence/imposter syndrome. Honestly, confidence is the one definitive things that people invest in independent schooling for!
There is a concentration of children in Whitgift & Trintiy for this reason. And, despite all of the stereotypes, many full fee-paying, academic children, are in these schools over more academic schools that they have been awarded places in, because of the amount of black, not multi-ethnic, but black children already there!
Many posters see 'diversity' but that's such a lazy term. Just in the same way that I would not expect Irish, Scottish and German people to have the same culture, I don't expect African, Carribean, Chinese and Indian people to either. Yes, many London schools are diverse, but most, across the board, are willfully underrepresented when it comes to black children.
Despite the stereotypical assumptions made for the reasons why, for many black parents (not all, but the vast majority) that are in a position to independently educate their children, unless they can get their children into a handful of schools, many opt for state schooling instead to save their children the trauma of being so drastically 'othered'.
And, unfortunately, many children of mixed (black) heritage will also feel this way in adulthood but perhaps grown by parents that won't have this initial insight until the children are able t later relay understand, articulate and relay this information back to them - usually well into adulthood.