One of the things that has come out of some of the workshops I run is that even "design" is so broad that "designers" can't know everything about design, either

One of the things that has come out of some of the workshops I run is that even "design" is so broad that "designers" can't know everything about design, either; we have to specialize into visual design, testing, print, web, etc. What is important is having the theoretical/academic basis for all of them, because it can greatly inform your decisions about your work and let you pull from a wider swath of experience. I don't think you should have to be a graphic designer in order to be a web developer. You probably shouldn't be (and I find myself increasingly disliking generalists). But you should understand the principles of design, and know what goes into making a graphic work, and how designers communicate and the vocabulary, just as you should know the same sort of things on the back-end for the operating system, hardware and network environment your software is deployed on.