For the background, see the post: Classification of triply transitive finite groups
Thanks to the classification of finite simple groups (CFSG), we know that if $G$ is a finite $6$-transitive groups, then $G=S_n$ or $A_n$.
Question: Is there an easy proof of this result (i.e. without using CFSG) ?
I'm also interesting by such a proof for $k$-transitive groups with $k$ sufficiently large.
If such proof doesn't exist (yet), are there people working on ?
I hope such an easy proof (will) exist.