I think most people need something to do for good well-being. It doesn't have to be paid work, or any kind of work, but it should feel meaningful in one way or another. Usefulness (whatever that means) might correlate with that but it's likely not the only requirement.
I've had times in my life when I didn't really need the feeling of being overly productive in terms of work or academically and when unproductive days (in the traditional sense) didn't necessarily bother me very much because I had hobbies that genuinely interested me and gave me a good feeling. Passion is an overused buzzword, but I think if you've genuinely got a passion for something (and it doesn't necessarily matter that much what it is), that can improve your well-being even if it's not something productive in the sense most people understand the word.
If you don't have something like that, though, it's much easier to fill the extra time with something meaningless, and I think that negatively affects well-being.