One thing that might help is to think about this advice as saying you need to develop new skills rather than being about the one true way to do mathematics. That is, you have a lot of experience and skill at learning math systematically from the ground up like you did in your classes, but that's only one skill out of many that are necessary to be an effective researcher. You also need to hone your problem solving skills, you need to learn how to identify appropriate interesting problems and research programs, you need to learn how to efficiently wade through the literature to find what you need, etc. You have less practice at these things, so naturally you're not as good at them and so it's going to feel uncomfortable, but that's normal when you're learning new skills.

In other words, you're focusing too much on being well-rounded in terms of subject matter, but you're missing out on being well-rounded in terms of your skills.

That said, I do think it's worth following your interests, provided you still have enough time for your main research program. There's more time in a day than my brain can handle working hard on my main research program, and the more of the rest of the time that was spent learning other mathematics, filling in gaps in my background, or working on a side project, rather than say playing 2048, the better. So keep learning broadly and deeply, but don't let it eat into the time you're getting your main work done.