The fact that Sylow p-subgroups of a finite group are always conjugate is one way to prove that normal implies characteristic for a Sylow p-subgroup. (So if a group has simple subgroups of index 100 which generate it, and no normal subgroup of order 25, the group itself is simple. Hence, the Higman-Sims group is simple because the Mathieu group $M_{22}$ is simple. This is done in Wilson's "The Finite Simple Groups".) Two consequences of this are that if $P$ is a Sylow p-subgroup of a finite group $G$ and $K$ is a subgroup satisfying $N_{G}(P) \leq K \leq G$, then $[K:N_{G}(P)] \equiv [G:K] \equiv 1 \mod{p}$ and $K$ is self-normalizing in $G$. In particular, the maximal subgroups of $G$ containing $N_{G}(P)$ are constrained by these results.
The cyclicity of groups of order 15 is more than just a toy example, since the cyclicity of groups of order 299 = 13*23 (which is provable the same way) is used in Thompson's original proof of the simplicity of the Conway group $Co_{1}$. (This proof also gives an example of the use of the Frattini argument.)
If you want to prove the Burnside $p^{a}q^{b}$-Theorem, you need to exploit the existence of Sylow subgroups. This is one of the few commonalities of the character-theoretic and character-free proofs of the theorem. Via character theory, the basic group-theoretic result is that a finite group with a conjugacy class whose size is a power of a prime cannot be simple -- but you can only get a conjugacy class of size equal to a power of a prime in a group of order $p^{a}q^{b}$ by choosing a nontrivial central element of a Sylow subgroup (unless you made a bad choice and it's in the center of the whole group, in which case nonsimplicity of the group is immediate unless the group is cyclic of prime order).
Eschewing character theory, Sylow subgroups are indispensable, whether you use the Glauberman $ZJ$-theorem or any other local-analytic tools to do the heavy lifting in the proof. They are also essential even for much lighter lifting which happens in these proofs.
When using the transfer to prove a finite group satisfying certain conditions is not perfect, it's good to have a subgroup from which this fact is visible. It is good to have a subgroup $H$ such that one knows $\phi : G \to A$ is nontrivial because its restriction to $H$ is nontrivial. If p is a prime dividing $| \phi(G) |$, then any subgroup whose index is a nonmultiple of p will work. A Sylow p-subgroup of $G$ fits the bill perfectly, and often comes with a fair amount of information about its own structure, to boot.
It is possible to build off of the Burnside $p^{a}q^{b}$-Theorem to prove the that the existence of Sylow systems characterizes finite solvable groups. Sylow system normalizers are all conjugate in a finite solvable group, and these facts form the starting point of the theory of finite solvable groups (which is substantial in its own right, as one can read in "Finite Soluble Groups", by Doerk and Hawkes).