Finite simple groups and applications of their classification 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?

 A: 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.
