I've spent literal hours on my island while doing roulettes. Just chilling. Enjoying the atmosphere without getting FC recruitment spam or limsa piss chat or tells about someone giving away gil because they're leaving the game. It's MY space. Being able to come back to the calm of my island was rejuvenating after hours of unsuccessful pf Savage prog attempts. It is a joy to go tearing around my island on Fatter Cat with its rushed pitter patter, hilariously launching myself off ledges to see what kind of distance I can get. I love going down to the beach on a clear day/night and just enjoying the view. If I bring all my mats with me, I can even spend time crafting there without anyone bothering me (or standing on me). I do a lot of content with other people, and the island gives me a change of pace for when I want a break but would like to keep playing.
If you don't view the island purely as a set of daily tasks to be performed, there are plenty of reasons to hang out there for more than 15 minutes.
Personally, I see the Islands coming from the same place as the patch release extension from 3.5 months to 4. Self-care and occasionally slowing down is important, and it's clearly important to Yoshi-P and the Devs to enact those values a bit through the game.
And as for the rest cycles, I think it's just intended to be another element of the workshop minigame, where you have to plan for them in your production efforts. (That, or it's a keen reminder to respect our robot workers so you survive their eventual uprising.)
