Is there a conceptual proof that $A_n$ is simple for $n \ge 5$, or that it is the only nontrivial normal subgroup of $S_n$?

Question

All the proofs I’ve seen of this are very combinatorial and at least for me not very memorable. Is there a cleaner or more conceptual proof? For instance, I wondered if representation theory might yield something easier. $S_n$ is such a ubiquitous group, that I imagine that looking at an appropriate action and employing techniques from the category in which the action lives, we might find a cleaner proof.