I've done multipule playthroughs and was able to do Cole's Odd thing option, (even with Deluxe & Frostback DLCs in place) on my 360. Recently I've run into problems where one specific lady at the gate or it vanished entirely on my Xbox One. I could still look for every other option after I made cole more like a spirit (sided with mages). On my playthrough with making Cole a Human (sided with templars) however, I lost it completely, it even vanished from Cole's conversation options. The only difference, other than the systems, seems to be the fact i have Descent & Trespasser installed. Bcuz I was still able to do the odd things on my 360 after I made him more human.
You could try uninstalling Trespasser (if you have it) and playing the game without but it would remove all the new bonuses from the game, seems like that odd things may or may not be as important?
I was thinking of trying to track it all down on a new playthrough, but I currently have two going and they're both bugged and probably won't start a new one for some time.