I say many (most?) mathematicians with a thriving research careers completely ignore large parts of mathematics in their work. Probably, they don't even remember what they learned in some of their introductory graduate courses, unless they teach them, and would be unable to pass some comprehensive PhD exams without preparation. What you don't use you forget.
Disliking some parts of mathematics is a way of finding what you really enjoy, a completely natural process. Being broadly educated helps, as long as it does not interfere with research. Learning and doing math are somewhat different activities. One cannot do math without learning some. On the other hand, it is possible to enjoy learning so much that you never actually do anything. There has to be a balance.
In particular, most math research can surely be done without category theory. If you ever need to learn what is, say, a colimit, just read Wikipedia, and follow the references there.
Short term, grad students should focus on finding the kind of math they enjoy doing, and also on passing their exams.
Personally, I revere broadly educated mathematicians, and I strive to become one. Is it a must for a successful career? Not really.