I've got some thoughts--we're pretty minimalist in our schooling--but I'm only teaching one so far, and he's only in first grade. My sense is that there are a LOT of a variables that could affect how minimalist one could be, and I've started to type this several times only to end up with an essay, so I'm just going to list what we do and you can ask if you want me to expand on anything, m'kay?
1. I don't have a separate schoolroom; instead we use the dining room table and keep everything we need for formal schooling nearby in or on a waist-high two-door cupboard. Yes, this means I have to put everything away before we eat. However, a) I would need to put everything away anyway, at least in theory, and b) we're only talking about a few books and a very limited set of manipulatives.
2. Note my emphasis on formal schooling above. This means basically math and handwriting. Other books and crafts are located elsewhere in the house; I don't really think of them as part of school because the kids access them all throughout the day. So the core of what's on/in the cupboard: a binder holding our math curriculum, a notebook for copywork, a mason jar full of rods and one full of glass beads for counting, a ruler, a whiteboard and some markers, a bucket of things to write with. . . that's really about it. I've got some more stuff stored there, mostly books, but it could really be anywhere in the house because we don't make daily use of it. I could post a picture if you think it would be helpful.
3. We use fairly minimalist curricula, especially in that we follow Charlotte Mason (more or less). Happy to give more info on what we use, but that's such a personal and philosophic choice that I'm hesitant to dwell on it. That said, our choice of curricula probably allows us to be minimalist more than anything else.
4. Organizing books: A big chunk of our Charlotte Mason-based material is on the Kindle. Honestly, though, it wouldn't make a dent in the number of books we have about if we used hard copies instead. My suggestion for book organization is baskets. You can organize them by theme (a science basket, a fiction basket, etc.) but really I just stick everything in together and the kids pull out what they like.
5. Organizing crafts: Ha. I'll get back to you on that if and when we ever hit on a system that works. Right now my 6yo and 2yo each have a craft desk, and I have a small bookshelf nearby that holds the materials ready to use. I'm trying to adopt a Montessori-get-them-to-put-stuff-away approach but we're not there yet.
One final point, now that I've re-read your post: I notice what you said about "paradox of choice" issues. I suffered from those myself, in a truly embarrassing way (at one point I described myself, with no irony whatsoever, as tending towards a Waldorf-inflected classical unschooling model--and to make it even worse I think my son was four when I said this). What really, really helped, not just from an organizing standpoint, but from so many even more important ones, was to determine my fundamental philosophy of child-raising and education. Once I had a clear goal and method in mind, it became much easier to resist curricula and materials.