When extending homology from chain complexes to the polyhedron of a simplicial complex, we use the simplicial approximations to prove invariance of homology. This eventually leads us to show that homotopic maps induce the same maps on the homologies of triangulable spaces (in simplicial homology).
My question is: what benefits does this stronger definition offer, rather than using a simplicial map that is homotopic to the original map?
A similar question is asked here: Definition of simplicial approximation
However, I am interested in why the condition on the Star is so powerful (i.e. what does the definition mean intuitively, and is there something that it can help prove that the weaker condition cannot?)