“Instead, I’d hold out for a top 10 program at a minimum, preferably top 5. And if that comes thru for him great. If not, I’d look to a MA/MS to continue to build resume, and reapply.”
I agree. Has he considered applying for Cambridge Part iii maths (https://www.maths.cam.ac.uk/postgrad/part-iii/prospective.html)? It seems to me like that would be helpful preparation after rushing through an undergrad degree. And in common with many UK institutions they have something of a soft spot for Berkeley.