# The significance of CW-complexes in homotopy theory

I try to understand the significance of CW-complexes in homotopy theory, in particular with respect to the classical models structure on $\mathbf{Top}$. Why do we chose Serre cofibrations for the classical model structure instead of Hurewitz cofibration?

About every source on homotopy theory mentions that the category of (spaces with homotopy type of) CW-complexes is the place to do homotopy theory, but I haven't found any source that explains why (except for comments like: if you read the last 200 pages, it should be clear).

Two nice properties I've singled out thus far are that CW-pairs satisfy the HEP, and the Whitehead theorem. But I doubt that this is the whole story.