The elephant in the room is moving out of London for the schools - less "ethnic" kids
The guardians own articles on property make a point of highlighting "good" schools when they cover an area?
Parents good intentions can often go out of the window when it comes to their children - I recall my mum commenting that if we had stayed in Birmingham she woudl have tried to get me into King Edward VII via family connections.
King Edward VII being the top selective school in the UK
Best determinant of that are the parents themselves. So private schools end up being essentially a sieve that checks parents. In the US parents move to the suburbs with high local taxes that fund good schools for the same purpose.