One of the things I like most about some of the quests is forcing you to build a very different party, but still within the same difficulty range. "Only Drawbacks" is the biggest example of this, but "Use Only Warriors" is one as well. I don't get the same feeling from "No deaths", "low level", or "1hp"(1hp forces a very different build but after the early levels gets much more difficult than the others) AotA added a whole bunch of new items, but only a few new scenarios. One way to get more out of that content would be to add quests to every existing map that say "Use only items from AotA". My expectation is that this won't be harder overall than the 1hp challenges. If you're worried about people not having enough AotA items to complete these, the quests could provide AotA chests instead of normal chests. The early missions should be possible with very few items, and they'd roll into more items later. I haven't checked an AotA item list but I assume there's at least one new card in every slot. If that's a problem, or you feel that some levels just won't be beatable with the cards available, you could allow non-AotA commons, but that muddies the quest a bit. For someone new to AotA, this is almost a way to experience the campaign again(assuming you unlocked the ability to do quests) - since your item pool is smaller, and you get rare chests for beating them. It would of course be great if these were free, but I'd think something on the order of 100-200 pizza would be reasonable - of course if I paid pizza that might imply more testing to verify that they could be beaten. I understand development resources are limited, but this seems like something that would be fairly easy to add (not much UI etc) while adding a fair amount of singleplayer content.
If they have a system that can already differentiate drawback card items, I'm pretty sure they should be able to set up AotA only quests (since they are already filterable). I'll third this suggestion.
There is also allready a button that display AotA items only (or basic set only, or both). And I wouldn't mind more quest, got only one left (drawback at Black Plume) but gotta first grind for some uncommon I need 5 more copies of