After completion of classification of finite simple groups (CFSG), many new interesting results came as its applications. For example, if $G$ is a finite group and if $p$ is a prime which divides $|G|$ but not the degree of any irreducible complex representation of $G$ then Sylow-$p$ subgroup of $G$ is abelian and normal. The proof of this theorem (of Ito) depends on the CFSG.
However many mathematicians doubtfully say that CFSG has been completed. Thus if a theorem is proved using CFSG, it is worth to ask if it can be proved without using CFSG.
Then my question is following:
Is (are) there any theorem(s) which were initially proved using CFSG and later proved without CFSG?
This is from the mathematical review article on the paper
S. Cohen, M. Fried, Lenstra's proof of the Carlitz-Wan conjecture on exceptional polynomials: an elementary version. Finite Fields Appl. 1 (1995), no. 3, 372–375.
Let $q$ be a power of a prime number $p$. A separable polynomial $f(X)\in F_q[X]$ is called an exceptional polynomial (E.P.) if the polynomial $(f(X)−f(Y))/(X−Y)$ has no absolutely irreducible factors in $F_q[X,Y]$. Carlitz conjectured that for odd $q$, there is no E.P. of even degree over $F_q$. Later on, Wan stated the following stronger conjecture: If $(n,q−1)\ne 1$, there is no exceptional polynomial of degree $n$ over $F_q$. Using covering theory and the classification of finite simple groups,
M. D. Fried, R. M. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), no. 1-3, 157–225,
gave a proof of the Carlitz conjecture on E.P. In this paper, the authors give a proof of Wan's conjecture. Their proof, inspired as they say by ideas of Lenstra, is quite elementary.