[isotypic]

One application I like is the use of the Seifert-van Kampen theorem to prove that the fundamental group of the circle (S^1) is isomorphic to Z. While category theory is not strictly needed to prove this (you can compute pi_1(S^1) using R as a cover in a way that is purely topological, see Hatcher "Algebraic Topology"), if one states the Seifert-van Kampen theorem for groupoids (this uses category theory through the notion of a universal property/pushout) one can compute pi_1(S^1) largely algebraically just from the universal property - in fact you can go through the whole proof without mentioning a homotopy once (see tom Dieck "Algebraic Topology" section 2.7).

This might not meet your criterion exactly, as one can extract a more topological proof and relegate the category theory to a non-essential role, but this requires some more effort and is a harder proof. So I do think it still illustrates that the category theoretic approach does add something beyond just a common language.

[sesm]

As far as I understand, fundamental groups were defined by Poincare in 1895. And functors in category theory are a generalisation of this idea (i.e. proving something for fundamental groups and then relating this back to topological spaces). So your example sounds backwards to me.