The theorem I am referring to in the title is this:

Theorem. If $p$ is a prime and $n$ is an integer with $n \geq p+3$, then the only primitive permutation groups on $n$ points containing a $p$-cycle are $A_{n}$ and $S_{n}$.

Without using the Classification of Finite Simple Groups, it seems hopeless to extend this to a statement detailing when it's possible to have a $p$-cycle in a primitive permutation group on $p+1$ points, since the Mathieu groups $M_{11}$, $M_{12}$ and $M_{24}$ arise this way when $p = 11$ or $23$.

The intermediate case is when $n=p+2$, and here there is a readily available family of examples: If $p = 2^{q}-1$ is a Mersenne prime, then the group $SL(2,2^{q})$ is primitive on the $2^{q}+1$ points of the projective line over $\mathbb{F_{2^{q}}}$. It contains $(2^{q}-1)$-cycles as the only elements fixing exactly 2 points. These groups can also be enlarged by adjoining field automorphisms, though the primality of $q$ means that the only larger group obtainable this way is the whole $\Sigma L(2,2^{q}) = SL(2,2^{q}):q$.

I have read that Burnside was able to prove that the only finite simple groups of even order in which every element has order 2 or odd order are $SL(2,2^{m})$ for some $m \geq 2$. My question is: Are methods not much stronger than the methods Burnside used to prove that result usable to prove no groups aside from those already mentioned act primitively on $p+2$ points and contain $p$-cycles?