If $x \in S_n$ with $x \neq 1$, $n \neq 4$, then there exists $y \in S_n$ with $S_n = \langle x, y \rangle$

Question

We all know that $S_n$ is $2$-generated group and I found an article where the author's main goal is giving a proof about this theorem:

If $x$ is a non-identity element of $S_n$ and $n \neq 4$, then there exists $y \in S_n$ such that $\langle x, y \rangle = S_n$.

Now, I'm looking for an alternative proof of this fact. What I'm seeking is not a self-producted proof, but another proof published by some mathematician in some book or article. Does anybody know something about this?

(The article that I'm referring to is written by I.M. Isaacs and Zieschang, published in 1995. But I found an 1940 paper which talks about this theorem, so I think there could be another proof of it.)

I report here the main steps of the proof that I found:

1. Split the demonstrations into cases
2. For each cases, choose a permutation $y$ such that the group $G= \langle x, y \rangle$ is transitive, primitive and contains a transposition
3. Use Jordan's Theorem (If $G$ is a primitive subgroup of $S_n$ and it contains a transposition, then $G = S_n$) to conclude.