For Cambridge it doesn't actually make a difference to admissions chances, but the department advises taking mechanics anyway. It's absolutely not a hard requirement, and probably more than half of the students on the course haven't done further mech, but there is some reasoning for it.
Firstly, to get in to the maths course you need to pass the STEP exams. STEP has three sections, Pure, Mechanics and Statistics. The pure section only require the core further maths stuff, so doing more pure doesn't help, and the stats requires very little material, I taught it to myself over a week or so. Most of the mechanics section on the other hand is next to impossible if you haven't done at least some further mechanics. Of course, you only answer about half the questions on a STEP paper, so you can choose to just skip mechanics altogether, but it is still a disadvantage to have less choice.
Secondly, the first year of the Cambridge maths course assumes you've done all of the further mechanics available. Dynamics and Relativity is basically impossible if you haven't. There are catchup lectures in the first term, but they are very fast, and typically poorly lectured.
tl;dr mechanics isn't required by Cambridge, but is helpful. If your son prefers stats then he should take it. At the end of the day getting that A* is the most important thing.