So, I've been thinking about this, and I'm going to say that it's not a problem, it just looks like one.
All Controllers are Peripherals. It's true, the former is a subset of the latter, and if we wanted to do a synonym, the appropriate thing to do would be to do a bunch of retagging and put the synonym in that direction. But I don't think that's a particularly good idea either, because the vast majority of our peripherals questions are in fact about Controllers. For the same reason that questions on Cooking about Chicken shouldn't be retagged Poultry. Most of them are specifically about the subset, and the subset is big enough to stand on it's own.
As to the question of 'what is a peripheral that's not a controller?' Well, admittedly, we haven't gotten many questions that fit the bill, but realistically, any questions regarding memory cards, external hard drives, network adapters, alternate media accessories, voice chat headsets, remote controls, and more would fall into the category.
A much simplified version of Steve's flowchart could then be: Is it a device I plug into my game console/computer? If Yes, then it's a Peripheral. Do I use it to control my games? If Yes, then it's also a Controller, and may not need the less specific Peripheral tag as well..
Whether a question needs one or both tags is ultimately, up to the asker and the community, keeping tag limits in mind.
In other words: Status-quo is fine, but we as a community should probably keep an eye out for mistagged questions.