A great example of the usefulness/necessity of bicategories is the theory of parametrized duality in May-Sigurdsson's book Parametrized homotopy theory, and the associated notion of trace which Kate Ponto and I have been working on.

Spanier-Whitehead duality can be described in purely point-set topological language, but it becomes much clearer when phrased as duality in the symmetric monoidal stable homotopy category. The Lefschetz fixed-point theorem then becomes simply a consequence of the functoriality of homology.

The most useful parametrized version of Spanier-Whitehead duality, called Costenoble-Waner duality, can also be described in purely topological language, but as May and Sigurdsson realized, it becomes much clearer when phrased using adjunctions in a bicategory of parametrized spectra. In particular, certain dualities which are quite tricky to construct explicitly now follow from purely formal considerations. More refined fixed-point theorems involving the Reidemeister trace and the Nielsen number also follow from formal considerations, cf. here.

I realize that bicategories are a bit low-dimensional, as higher-dimensional categories go, but there are some indications in May-Sigurdsson of a need to go at least one level up to some sort of tricategorical structure. And of course all of this is happening homotopically, so really we are at $(\infty,2)$-categories or $(\infty,3)$-categories.