Lots of things are groupoids, but many are not groups. There is a theory of groupoids, and if you don't acknowledge groupoids, they won't let you use their theory =)

The fundamental group(oid) example is really good. What happens if you want to do Van Kampen's theorem on a pair of sets whose intersection is not connected? There's an answer but you have to use fundamental groupoids.

Also, categories are sometimes useful for isolating data. For example, you can replace the usual notion of a local system with a "representation of the fundamental groupoid" and doing so lets you think of a local system the way you already wanted to in your heart: as a collection of operators on fibers coming from "going around the bad points".

Here's a stupid riddle for everyone: what's another word for a "monoidoid"?