I think a late January birthday is quite young and your son may struggle socially just because BS kids lean old, particularly at schools that have a large contingent of kids repeating ninth grade --my sense is that this is fairly common for students who went to junior boarding schools, K-9 schools, and international students who want to bolster their English skills.
Between students repeating a grade, redshirt kids, and the variety of kindergarten starting ages across the US, there can be up to a 2 year gap between the youngest students in a BS class and the oldest. However, I think the average age for ninth graders starting BS is probably closer to 15 than to 13. And the latest that I’ve seen kids turning 14 is Dec 31st (some school districts do kindergarten entry on the calendar year instead of the school year, but most currently demand that kids turn 5 by the first day of school).
My D24 with a winter birthday has classmates who are exactly 1 year older than her and exactly 1 year younger. So her freshman year, she turned 15 in the winter. She had quite a few classmates who turned 16 in the fall/winter and a couple of friends who turned 14 at the same time, but those were rarer. No one turned 14 after the end of the year.