Is there a gap in the standard treatment of simplicial homology? On MO, Daniel Moskovich has this to say about the Hauptvermutung:

The Hauptvermutung is so obvious that it gets taken for granted everywhere, and most of us learn algebraic topology without ever noticing this huge gap in its foundations (of the text-book standard simplicial approach). It is implicit every time one states that a homotopy invariant of a simplicial complex, such as simplicial homology, is in fact a homotopy invariant of a polyhedron.

I have to admit I find this statement mystifying. We recently set up the theory of simplicial homology in lecture and I do not see anywhere that the Hauptvermutung needs to be assumed to show that simplicial homology is a homotopy invariant. Doesn't this follow once you have simplicial approximation and you also know that simplicial maps which are homotopic induce chain-homotopic maps on simplicial chains? 

One doesn't need to explicitly compare  with singular homology to get homotopy invariance (although that is certainly one way to do it), and one certainly doesn't need the Hauptvermutung (thankfully, since it is false in general).  Rather, as you say, one can simplicially approximate a continuous map between simplicial complexes, and thus make simplicial homology a functor on the category of simplicial complexes and continuous maps (which, as you observe, will then factor through the homotopy category).  Just thinking it over briefly, it seems to me that the hardest step in this approach will be to show that the induced map on homology really is independent of the choice of a  simplicial approximation. Presumably this was covered in the lectures you are attending.  (In fact, I guess this amounts to the fact that homotopic simplicial maps induce the same map on homology, which you say you have proved.)   
