Question

By a Theorem of R. Guralnick and A. Lucchini (see MR1015993 and MR 1023965),(which does require the classification of finite simpe simple groups) the minimum number of generators for a finite group $G$ can exceed by at most one the maximum (over it all its Sylow subgroup subgroups $P$ of P$) f the minimum number of generators of $P$. It follows that the value of $D(G)$ is between $d(H)$ and $d(H)+1$ for some $p$-subgroup $H$ of $G.$ Thus for a finite simple group $S,$ the question does essentially come down to bounding the minimum number of generators of subgroups of $S$ of prime power order, as was suggested might be the case in some comments. Th The sectional $p$-rank of a finite group $G$ is the maximum number of generators of any section of $G$ which is a $p$-group (a section of $G$ is a group of the form $H/K$ where $H$ is a subgroup of $G$ an $K \lhd H ).$ Hence if we define the sectional rank of $G$ to be the maximum of the minimum number of generators of an Abelian section of $G$, and denote it by $ar(G),$ then we see that for any finite group $G$, simple or not, we have $ ar(G) \leq D(G) \leq ar(G)+1.$