Any simple group of order $60$ is isomorphic to $A_5$

It seems to me that there is a significant overlap with the standard proofs, e.g., see here. "Proof that there is only one simple group of order sixty, isomorphic to the alternating group of degree five": The key idea is to prove that $G$ has a subgroup of index five. After that, we use the fact that $A_5$ is simple to complete the proof. Again, Burnside is used. For another idea to use a stronger version of Sylow see here:

About the proof that a simple group of order 60 is isomorphic to A5

So to your question Has anyone seen this proof before? Yes, up to minor variations, which are always possible. In general, this theorem has been treated so often, that almost everything has been said. I am not sure that it is interesting to come up with yet another variation. On the other hand, it is certainly useful to write this up for oneself, and for further topics in group theory.