This question is a sort of a follow-up to these two: reference on classfication of multiply transitive permutation groups and Multiply transitive groups, continued

The question is simply: is it true that the set of $n,$ for which any doubly transitive subgroup of $S_n$ is one $S_n$ or $A_n$ is of density one (since any doubly transitive group is either affine or almost simple, this seems clear, but I might be missing something silly).