I think many people share my sentiment that existing proofs of Sylow's theorems, although very clever and amusing, are not "conceptual" or "explanatory"; it seems to me like a massive coincidence that every finite group has a Sylow $p$-subgroup for every prime $p$.
I have thought about this question only very superficially, so I don't have a lot to offer, but here goes:
Suppose we didn't know that every finite group has a Sylow $p$-subgroup for every prime $p$, but we did know this for finite simple groups.
Now, I want to imagine how an inductive proof of the existence of Sylow subgroups might go.
Hypothetical Premise: We know that every finite simple group has a Sylow $p$-subgroup for every prime $p$. This is also the base case of our inductive proof.
Inductive Step: Let $G$ be a finite group. Let $M\triangleleft G$ be a maximal normal subgroup. Since $|M|<|G|$, we apply the inductive hypothesis to see that $M$ has a Sylow $p$-subgroup. Since $G/M$ is simple, we know that $G/M$ has a Sylow $p$-subgroup.
Now, given this, we wish to construct a Sylow $p$-subgroup of $G$. How could we do this?