The notion of an elementary collapses collapse is the key construction in the definition of a simple homotopy-equivalence. Simple homotopy type is a refinement of homotopy type, and it has many uses.
One example: The s-cobordism theorem is the main structure theorem for high-dimensional manifolds, and it has to do with the question of when two simple-homotopy equivalent manifolds (technically s-cobordant manifolds) are diffeomorphic.
You might want to take a look at Marshall Cohen's introduction to simple homotopy theory textbook. One of the key examples there is that simple homotopy type is the same as homeomorphism type for lens spaces, but homotopy type is a different, weaker relation.
Whitehead torsion and Reidemeister torsion are simple homotopy invariants. In a sense you can think of simple homotopy theory as a space-level analogue of elementary row and column operations on a chain complex. So it gives you a sense for why it should somehow be more relevant to forming a bridge between topology and algebraic constructions.
As for Bing's house, I think the answer is kind of simple. Where would you start the collapse?