(I only just saw this one year on!) I find your question very strange. Grothendieck gives a simple categorical formulation of a situation that encompasses the three main examples of Galois theoretic machines. That means he shows what makes things really tick... isn't that good enough for you! He does this with the clearly stated aim of developing a fundamental group for schemes, and the theory gives that and a lot more. If you go to the slightly wider results on the fundamental groupoid of categories of locally finite sheaves, that is a first step towards his Pursuing Stacks, the letters to Larry Breen, and enroute for his Longue Marche.
In another direction it provides a first step towards the Joyal-Tierney theory of locales etc. and their relation with toposes. It provides a background for all of Jacob Lurie's work on higher toposes, and I could go on with fundamental groups of toposes, homotopy theory of toposes. SGA1 is the key for understanding a large part of modern mathematics.
Grothendieck's methodology was always to seek the clarity that came from abstraction and generalisation. His aim was not only to solve problems (say in algebraic geometry) but to understand as fully as possible their solution and why they worked.