I think the comments here converge toward the answer "no" to your basic question. While some very special small primes are treated in a few textbooks (using what seem to be ad hoc methods), it's much more natural to treat the entire family of groups at once in the wider context of the classification of finite simple groups. Here there should be some general methods in play, e.g., BN-pair structure, structure of "local" subgroups such as the normalizers of Sylow subgroups, centralizers of involutions, etc. Not to mention many techniques from character theory.

I'm not aware of any treatment of this special family of rank 1 groups in a form suitable even for an honors course. The two books Danny Gorenstein wrote on finite groups, especially the second one published in 1982 when he and others felt the classification was complete, illustrate the difficulty at that time in organizing the subject in textbook form. In that later book the special result you want is embedded in a substantial treatment involving doubly transitive permutation groups, Zassenhaus groups, etc. I don't think it's feasible to extract from that textbook version an efficient proof for your purpose without bypassing the important underlying general ideas, though it's always interesting to see what can be written down in a more elementary and self-contained way.

Anyway, the problem here is to work out in a very limited case the "recognition" theorem for finite simple groups: how do you argue that an unknown simple group (here specified just by an order formula) is isomorphic to some known group?