I feel topologists should be eclectic and able to move across a number of models, to see which better aids understanding. Here is a part quotation from Einstein, which I feel also reflects the concerns of the questioner:

"What is essential and what is based only on the accidents of development?... Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. ....It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little..."

Simplicial sets have a very well developed theory, are "convenient" in many ways, but they have limitations. The results of the book Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical omega-groupoids (NAT) (pdf available) would not even have been conjectured simplicially, but the notion of multiple compositions of cubes led to their conjecture and proof. A search on "cubical" in mathoverflow gives more relevant information. To go back to the question, the above book gives a new outlook on structures related to CW-filtrations, and the border between homotopy and homology, using cubical sets (with connections).

See also this brief 2015 presentation on "A philosophy of modelling and computing homotopy types": aveiro.

January 5, 2016
Since products are referred to in other answers, I mention that the isomorphism $$C_*(X_*) \otimes C_*(Y_*) \cong C_*(X_* \otimes Y_*)$$ in the cellular case is extended in this case to an isomorphism $$\Pi(X_*) \otimes \Pi(Y_*) \cong \Pi(X_* \otimes Y_*) $$ in the above NAT book. Here $\Pi$ is a homotopically defined functor on filtered spaces with values in **crossed complexes**: this functor contains information on relative homotopy groups $$\pi_n(X_n, X_{n-1},x), x \in X_0, n \geqslant 2,$$ and on the operation of fundamental group(oid)s on these. Proofs use cubical methods, relying on the isomorphism $I^m_* \otimes I^n_8 \cong I^{m+n}_*$.