Ok, I think I figured the actual behaviour out. The initial quest completion shows 1/3, because one dungeon had to be looted to unlock that quest.
Looting one more dungeon seemingly doesn't do anything, because the counter gets updated to 1/3. Looting another one, pushes the counter to 2/3.
The previous message had me loot a single dungeon, check the quest progress directly afterwards and give up, thinking the quest isn't updating.
btw, the new quest "too useful hex cores" initially shows 0/500 progress, despite me being at 700ish when loading the save. It properly updates on looting the dungeons (I guess any expansion does that), but it would be nice if the progress is shown.