I do love diagrams, I love thinking about some problems in terms of diagrams, eg Feynman diagrams, some enumerative problems, knot theory etc. It can be kind of deep. I think they are good for when you can do some graphical manipulation to prove a statement, or help gain extra insight. Usually there is some monoidal category floating around.

Perhaps one area this could be useful is program simplification? It would be nice if I could take a more complicated thing like a Haskell program or function and then do some graphical manipulation to see how to rewrite it. Kind of reminds me of pointfree too. Sometimes that can lead to interesting insights, where you spot something more general or simpler. You are just focusing on the functions (==diagrams) rather than the strands leading in.