I am not sure that I know a really elementary proof of this fact, though I can imagine
ways to start reconstructing one (Sylow theory and, as you say, transfer seem
good places to start). But, in the end, you need a way to recognise this simple
group. Seeing later comments, five pages sounds reasonable. There is a Theorem  of M. Herzog (circa 1970) which (roughly speaking) characterizes finite groups of order less than $p^{3a}$ with a cyclic Sylow $p$-subgroup of order $p^{a}$, but this uses fairly sophisticated character theory,
and (including background) needs a lot more than five pages.