Just turning comments into an answer.

1. **Iwasawa's Criterion**. This will do $A_5$ (using the natural action), $A_n$ with $n> 5$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are [here][1].

2. **A list** of the type you asked for [does exist][2] thanks to Keith Conrad (it contains five proofs of the simplicity of $A_n$, for $n\geq 5$.)

3. See the comments for other suggested proofs.


  [1]: http://boolesrings.org/nickgill/2014/03/11/topics-in-group-theory/
  [2]: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/Ansimple.pdf