Modern algebraic topology, especially homological algebra, more or less requires category theory... intro textbooks such as Rotman's will contain primers on category theory for this reason.
I'd say Grothendieck's proofs of the Weil Conjectures is a good example. His proof uses etale cohomology and the definition of etale cohomoly uses Category Theory in a fundamental way. From the etale cohomoly wikipedia page https://en.wikipedia.org/wiki/%C3%89tale_cohomology
"For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to X. There is a rather minor set-theoretical problem here, since Et(X) is a "large" category: its objects do not form a set."
There's a lot of advanced math in that paragraph, but it should be clear that Category Theory is needed to define etale cohomology.
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.
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.
Putting topology aside, and recognizing that 'ease' is subjective, imo Moggi's use of monads to model the denotational semantics of I/O in lazy functional languages such as Haskell is a common textbook example; the creators of Haskell had tried many solutions that did not work in practice before monads cracked it open. Even now this solution is more widely adopted than the alternatives (streaming I/O, linear I/O types, etc) and Moggi's paper remains a classic.
