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The topological proof of the Euler polyhedron formula is a special case of the general proof that the alternating sum of Betti numbers equals the alternating sum of numbers of simplices, which follows from the Hopf trace formula proved after 1925, after Hopf had learned from Noether to consider homology as abelian groups, while Poincaré was considering Betti numbers and torsion coefficients just as numerical invariants, not as algebraic objects. I guess that it is hard to avoid thinking of chains and homology as abelian groups (or better vector spaces) if one wants to prove the Hopf trace formula, so it is not clear to me how Poincaré might have gotten at a topological proof of the Euler polyhedron formula. Also topological invariance of Betti numbers was not known at the time. (But perhaps there is a simpler argument just for the 2-sphere?)