One aspect of London's diversity is the diversity of parenting styles. I believe Word when she says her children have not been tutored, and I expect her to believe me when I say the same. (Though in fact dyslexic DD had some extra, very useful, help in English which went a long way to establishing some confidence.)
It would not surprise me if some Westminster children were tutored. Indeed me it would surprise me if some weren't. It would actually surprise me if there were a selective school in London, state or private, where some children weren't tutored. And there will be different reasons. Sometimes because children are stuggling, or absolutely need that B in Maths or English, sometimes because they were tutored to get in, and need tutoring to keep up. I remember one mother complaining that DDs school made it very difficult to compare your child's performance against that of classmates. She did not know which subjects to organise tutoring in. It turned out that she came from a system where class ranking mattered, and so she wanted her, very bright and lovely, child to be at the top of the class in all subjects, eventually graduating in first place. I think at that point her child was already being tutored in four subjects. Really not needed!
Tutoring teenagers without a specific purpose, however, can be a double aged sword. We have come across more than one teenage who relies on the tutor to do homework. Indeed, and this was a boy at a very famous school (not W), the tutor did the GCSE coursework. My children tell me they learn best by listening in class, which is a good learning habit. Some kids instead learn to rely on tutors instead to go through material, which must frustrate teachers.
OP sounds a couple of mothers I know, indeed I wonder if I might know her..... Its been strange. I know there is an element of this in MN but in the world of central London private schools, there seem to be ranks of hugely gifted children. DC never were. At times it has felt as if the parents have judged us on our ordinary public sector jobs and then their children have assumed the same attitude towards our children. (A bit tedious, and not dissimilar to the attitude towards career tutors which seems to exist here.)
What I have wondered though is what happens to these children. They are centre of attention. If something goes wrong, the school, school system, teacher, whoever, is blamed. (One mother even threatened to sue a sports coach for emotional damage to her child!) Some really are renassiance children with great music, sports and other extra curricular, on top of a very solid academic performance. But what happens after that. They may go to Cambridge. They may thrive there. But then what? I guess the assumption is they go on to the States for post-grad and then into a top law firm or somewhere like Goldman Sachs. But work demands other skills beyond academic success, and numbers, dictate that more than some will fall off their pedestals at some point on the journey. And even if they make it, should they ever pause to reflect, is this where they want to be. Do they have the wider skills to be a good spouse or a good parent. Do they have the hinterland, empathy and resiliance that they might need in the future.
I probably know/know of one too many banker wives who have discovered the nice house in Kensington is not worth sticking in the marriage that provides it!
This I admit is off-topic, and may not be relevent for OP and her DS. It also reflects other recent conversations. Not everyone is motivated by money or employment status. It is perfectly possible to be very very good at maths, or indeed anything else, and simply not be suited to working in organisations, including the political organisations that schools can be. DD did some career profiling which suggests she is very interested in caring for others (strange, she showed next to no interest in caring for her rabbit!) yet not interested in law, research or policy, business or anything artistic. That fine. Becoming a postman, if she wants to be a postman, is also fine.
My sense is that its worth finding the field that suits you, and then trying to identify roles within that field that match your skills. Then you have a good chance of both doing well and achieving job satisfaction. Maths and tutoring sound like a good fit for some, and the demand is there.