There is auto-exploration button to keep fighting foes automatically, and also a button to automatically advance areas. The unknown tab indeed requires some baby sitting. Sorry to disappoint!
Something buggy though. I've tried that button already and it didn't work. (Now I've tried it again and it works)
Not sure what's causing it or what prevents it from working.
Anyway: Now that I know that it CAN work, the game is far more fun than before.
Another bug I found: I have an item "Nature Warrior of Thunder", which simply locks the game after one battle. The next battle is visible, but the game does not react to inputs. I can't even change UI screens then.