This app was mentioned in 3 comments, with an average of 3.00 upvotes
Thanks for the recommendation. I have just found the original game thanks to an old screenshot, it was called "DungeonDiary"
https://play.google.com/store/apps/details?id=jp.windbellrrr.app.dungeondiary&hl=en
It's not very pretty or got much to it but it's about setting rules and watching them go. Also something something make your own waifu
https://play.google.com/store/apps/details?id=jp.windbellrrr.app.dungeondiary
Well, I've done a little bit of research.
DungeonDiary is probably the closest in spirit to what I'm looking for, at least as far as "set it up and go" works. You play as a moe little girl with a staggering 30 million gold debt to pay off, so time to do quests. The game works by setting "rules" that define what your character does, like "I will pick up [mushroom] below [B1F]" or "I will return to the floor entrance if I pass an area [5] times" and so on. You can watch the action or go do something else and get notified when there's an update, at least.
AutoRaid: RPG Manager is "set it" but no "forget it" here, but fights are short enough that it's okay. You have a variety of cards representing different races and different classes, and it's up to you to put together a cohesive team. Set up the party order to determine who'll take the brunt of the damage and spend keys (energy system) to take on stages. Online only, unfortunately.
Merchant kinda fits the bill, given the "set it and forget it" thing though the waiting is only a minute to a few where I am now. It's a lot more on the side of being an actively-played game than the LQs with their hours of waiting. But it helps I like the game anyway so that's always a plus.
That's all I got so far.