VEBott Not in London, and then outside London you don't have to be rich /elite/prestigious/whatever to ride. In fact have regretted that we didn't have time /resources to distract our West London princesses with a pony. All those elite pupils necking back the vodka on Wimbledon Common of a summer's night would think twice if they had to be up at dawn to muck out.
OP my DDs went to a west London school that describes itself as one of the foremost academic schools in the country so I suppose it would regard itself as elite. DD1 knew it was the right school for her, indeed she rejected SPGS in it's favour, and she did thrive there. DD2 made her decision based on sibling rivalry and it definitely wasn't right for her, compounded by the fact that she found herself in an extremely dysfunctional year. Doesn't matter how elite the school they can't guarantee that having selected on academic ability the pupils won't bring along their baggage and insecurities, and disrupt and manipulate the rest of the year.
She is now at a less elite school, the one I felt originally felt right for her, and couldn't have done better at AS level. Moreover she is with a peer group who are not having to deal with some "difficult characters" . She says she thinks the teaching is better, her old school was too focused on results whereas this school is focused on educating them and on encouraging them to fulfil her potential. The old school gets better results but that is because it gets the brightest most motivated pupils.
Visit the schools, it is intangible but you and DS will know which school feels right for him, if you trust your instincts regardless of league tables, elite ness etc.
We are very lucky in West London to have many schools that are in the top 100 in the country and anywhere else would be regarded as elite but because of the ridiculous levels of parental anxiety and competitiveness notional league tables crop up based on the most minute differences in results, facilities etc. ignore all that if you are going to make the best choice for your DC