Proof of CFSG assuming every simple group is two-generated

Question

It is well-known that one of the corollaries of the classification of finite simple groups (CFSG) is that every finite simple group can be generated by two elements. In a comment on an answer to an old question, Kevin O'Bryant mentioned that

A group theory guru once told me that this is almost equivalent to the classification, in that the classification could be tremendously simplified if this (every finite simple group can be generated by two elements) could be taken as a lemma.

How does this simplification go?

An ideal answer would of course be one that gives a full proof of the CFSG assuming the $2$-generation property as a hypothesis, but that seems like a lot to ask for (and probably still involves some tedious case-by-case analysis?), so perhaps more reasonable is to ask about the broad outline of such a simplification.

As a bonus question, is there anything specific about $2$-generation here, or would it suffice to know that there exists some $k \geq 2$ such that every finite simple group can be generated by $k$ elements in order to achieve (some) simplification of the proof of the CFSG?

Answer 1

I don't know who the "guru" being alluded to might have been. One obvious consequence of every finite non-Abelian simple group being two-generated is that for each finite simple group $G$, the outer automorphism group of $G$ has order less than $|G|$ (just consider where the pair of generators can be sent under an automorphism, and observe that $|{\rm Aut}(G)| \leq |G|(|G|-1).$

A priori, without knowing the precise structure of a putative finite simple group $G$ , we would know almost nothing about its outer automorphism group, and in general, there seems to be no real reason to expect a bound of much less than $|G|^{\log_{2}(|G|)}$ for the order of its automorphism group.

Since analysis of a maximal subgroup $M$ of a putative finite simple group $G$ often requires understanding of the outer automorphism group of the generalized Fitting subgroup $F^{\ast}(M)$, and since $F^{\ast}(M)$ is a central product of the nilpotent subgroup $F(M)$ with the quasisimple subnormal subgroups of $M$, any knowledge of the outer automorphism groups of quasi-simple groups is useful (and the outer automorphism group of a quasisimple group $L$ is the same as the outer automorphism group of its simple quotient $L/Z(L)$).

This is very far from a justification of the fact that there would be a dramatic simplification of the proof of CFSG, and I clearly can't know what that person had in mind (perhaps it was something deeper and much more subtle), but it does at least point towards some potential simplifications if the two-generation of simple groups could be taken for granted. However, I find it hard to imagine how it would be possible to take for granted that simple groups are two-generated without knowing an awful lot about them.