Thanks for sharing this proof! As someone who enjoys math but never got myself through enough Galois theory to finish the standard proof, it's fantastic to see a proof that's more elementary while still giving a sense of why the group structure is important.

Sperner lemma is very much an algebraic topology theorem. The ideas involved in it form the basis for the theory of simplicial homology, which in turn will lead you to general homology and cohomology theories.