Looking at your list OP, (which seems focused on central London Preps and Junior depts that are seen as 'top'.
For Girls
Bute House (Prep) needs to be on the list.
Glendower needs to be on the list for girls, as does Pembridge, Ken Prep too really.
Co-ed
Latymer Prep is very highly ranked
Thomas's schools are popular .
Eaton House , but not exactly central in Clapham.
Once WUS goes co-ed, they need to be on any top co-ed prep list also.
I think all private schools in London are diverse, racially and culturally (not financially! full bursary kids make up a very tiny minority in any school, the minority will be on any bursary help at all. The typical bursary will be very middle class households getting 10 to 50% help, )
All London schools are diverse with working class parents working 3 jobs to cover fees to A list celeb parents and some Aristo's, and everything in between.
I think the oligarchy, global elite tend to flock to central London schools only then off to Swiss or boarding school somewhere. They seem much less anxious about academic schools compared to say professional middle class parents , they can afford to be I guess.
I've found old money tends to be outside of London , in sleepy village schools then boarding.
Most parents at London private schools are typically professional, working long hours.