The Solvability-Assumption in the Schur-Zassenhaus Theorem could be dropped by the Feit-Thompsen Theorem, but (as mentioned in the wiki-article) it could also be dropped by using Schreier's conjecture.
But how exactly would it be used here? (I do not have access to the unpublished note of Ernst Witt mentioned in the wiki article).
It does not seem to be directly related, the only way I can think of is by somehow arguing with factors of some composition series, which have solvable outer automorphism group if we assume Schreier's conjecture. But I do not see how to derive solvability of at least one of the groups $N$ or $G/N$ mentioned in the Schur-Zassenhaus-Theorem?