UPDATE: I didn’t have problems with the bugged auto-play for quite a while, and then it happened twice yesterday when I was using it for Kingdom Challenges (first time I’ve used it for the challenges.)
For me it’s actually the other way around, in very rare cases I don’t get offered Auto Play Mode at all. The menu just doesn’t have the option until I complete the current battle and move on to the next one. It feels like whenever it happened I was unusually quick to open the menu, not sure though. Explore mode, maybe it’s related in some way.
Had this happen again. Second battle in explore run, no auto-play option available. Battle was launched and tabbed out for at least a minute, first interaction was attempt to launch auto-play, so it doesn’t seem to be related to opening the menu fast. Red storm was active from Firestarter talent of Slayer hero class.