Great topic, OP
the main thing to know IMO is STEP -Sixth Term Examination Paper referred to above. Taken at the same time as A Levels, incorporated into all (or almost all) Cambridge Maths offers. Imperial and several other places use it as one option but allow candidates to use results from other, easier, exams sat the previous autumn instead. Oxford don’t use it.
Cambridge write and mark the exam nationwide. They set the grade boundaries to control the size of their incoming class. Typically only about 50% make their offer.
I think some of this is down to a lack if preparation. These applicants are the best mathematicians in their schools. They can’t conceive that problems on A Level material (both Maths and FM) could be so difficult. They are caught short. But it is also true that STEP is very, very difficult.
Note - you can get a top grade without getting a top mark. It is scaled.
Practice is the best thing. Are you in England? Does DS know about the Advanced Maths Support Network? He is entitled to join. His school can help. There is also MumsNetter who works on STEP outreach at Cambridge. Her name is something like Pallandro, perhaps, and she is fantastic. Perhaps someone can help us get it right and we can put out a call on this thread.
Also if DS’s school is in a weaker socioeconomic area it might be paired with a Cambridge College for outreach purposes (obviously he can still apply anywhere). They may be able to provide STEP resources. The problem is, pupils at independent schools and in leafy suburbs have access to STEP tuition. That’s what he needs. It may seem early, but it isn’t if he is doing A levels.
DH attended one of the old colleges. He made some lifelong friends. Academically he made a slow start, not helped by having a personal tutor who would not be tolerated today. Anyway, he ended up with a Distinction at Part III (Cambridge version of MSc) and an invitation to do a funded PhD. He turned it down to go elsewhere and had a great career as a professor at major universities.
My perspective from being an Admissions tutor is that Oxbridge sometimes make mistakes. Even when they get it right, they turn down many applicants who could thrive and do them credit. Everyone who applies must do so with resilience. Cambridge Maths applicants above all. Best wishes to DS