As https://lotro-wiki.com/index.php/Misty_Mountains_Quests points out, it's supposed to be an auto-bestow, and I could swear it always popped up. Now, however, I took a toon all the way up north to Helegrod to post the Evil Men warning, then back down into the West over to the chest, and the quest hasn't popped up.
I ran a toon through there semi-recently, a few months ago maybe, and I got the quest. Maybe wander the path between the chest and Gloin's camp to see if it triggers in a certain spot?
Got it! Thanks for the advice. I'd wandered over "almost" all of it trying to find the quest the other night. Went back and realized I hadn't been near the ridge separating it from Northern Bruinen Source. As I got close to that, the quest popped.