Mrs Guy I have known several kids who have flourished following a change of school. I am sure you did not mean to, but it sounds as if you believe that a problem with a school implies a problem with a child.
A decade ago but we turned down a place at SPS for DS, mainly because DD took part in an extra curricular activity at SPS. I mentioned a friend of DS's to a couple of mother's who had Colet boys of the same age. The reaction of both was the same. "Oh he is not very bright". The boy was 10 and quite probably brighter than my son, and we did not want our son to be under the same sort of pressure. One mother was able to tell me exactly where her boy sat in the year group and his plans to work his way up. Another parent, who decided not to move their child from CC to SPS, told me of some quite shocking shenanigans to get children onto sports teams, with her quiet son losing out.
There is an element of extreme competitiveness in West London. It is a minority and some, I think, is driven by the importance of things like class-placement, leadership positions, school reputation, and extra-curricular when it comes to US applications. Some is cultural. One lovely Russian mother once asked me to help her understand the downplaying of a child's performance relative to other children in the class. It was so different to her own experience where it was really important to be top...in every subject.
DD later described the atmosphere in her EC activity as "toxic". The assumption seemed to be that if you was not at SPS or SPGS, you were not intelligent enough, and if you were not intelligent and not headed for a world top 5 University, you were not worth knowing. Good enough was absolutely not good enough. (DD was dyslexic so not in the competition anyway, though by the time she was offered a place for Westminster sixth form she was able to be bemused when others suggested that she had got in because Westminster was not nearly academic enough for SPGS girls to consider.) And shockingly parents, as well as their children, continued to be happy to offer opinions on other children's intelligence.
I am sure that it is quite possible to go through either SPS or SPGS without noticing this. And perhaps only obvious when it comes to the sharing out of sixth form leadership positions, which the US bound kids will need in a way that those aiming for UK Universities don't. Several people also told me that SPS was a far nicer school than CC.
I would though be unhappy if a DC in a super selective school was denied a chance to take double maths. DS was in the bottom double maths set for A level at Westminster, and took it mainly because he wanted to learn as much maths as possible at school before he started a degree course with a significant maths content. In his case something clicked and he has got better and better. Others, who were very much top of his year have faded. In a couple of cases we suspect the super performance up to A level was based on very hard work, and possibly quite a lot of tutoring. (We later discovered that one very bright East Asian friend of DS' went to a full-time tutorial college every holiday. Atypical perhaps, but perhaps an explanation of why DS has been able to find his academic feet at University in a way he did not at school.)
There is a lot of pressure on some kids from the West London catchment. Schools cannot prevent this, but there is a lot they can do to help ensure that competition within a school is downplayed, with education and personal development valued, rather than emphasis on getting the grades and the sports team success.