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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).

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Igor Rivin
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Yes. In the paper

P.J. Cameron, P.M. Neumann, and D.N. Teague, On the degrees of primitive permutation groups, Math. Z. 180 (1982), 141-149,

the stronger statement is proved that, for almost all $n$, the only primitive permutation groups of degree $n$ are $A_n$ and $S_n$. (By a fortunate coincidence I saw a reference to this in a paper I was reading a day or two ago.)

Derek Holt
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