Unfortunately, their reservations about admitting him will have nothing to do with his academic aptitude. Everyone who gets into a math PhD can do the math, so it’s not in question and becomes a pretty irrelevant point when comparing applicants at the same level. The most important metrics that any applicant is assessed by are things like emotional maturity, resilience, independence - that’s why they’re usually explicitly asked to rate these qualities in their letters. Having an aptitude for math and research isn’t going to matter if you’ll burn out and drop out, and burning out is all too common for younger people.
There are many applicants far better than your son (this applies to most candidates) who are also older - that puts him at a disadvantage. Talking to him about math doesn’t magically make him older nor does it fix any of the red flags associated with his age. Him living on campus and having done an internship doesn’t assuage them either. PhD programs definitely prefer older candidates because they historically have a lower risk of dropping out. There is a strong preference away from candidates under the age of 20.
They may think he’s the smartest kid they’ve ever seen and still decide to reject him so that he can reapply when he’s older.