I haven't been able to complete the Sturm Quest on my hunter, the legendary items I've been picking up off the floor that auto-decrypt have counted to 5/5, but I found an exotic cosmetic item in a drop from a strike that auto-decrypted and didn't count towards the number. I read in another thread that maybe purchasing the fated exotic from Xur would work, it didn't, still 0/1. (and I didn't get anything useful, bummer). I rolled a Titan to see if I could complete the quest with the exotic engram you get after (I think) completing the main campaign, I saved it until I had received Drang and been given the next quest step, where I found that decrypting the exotic engram at the cryptarch didn't count towards the quest either, nor did any legendary engrams (bright or otherwise) count towards the legendary decryption goal. I can report back if I get a non-cosmetic exotic drop from somewhere (A public event or strike I guess?) but rng hasn't been on my side so far, and I still feel like I'm jumping through hoops a bit here. TL;DR: Can't complete Sturm Quest, exotic engrams looted, bought or obtained as rewards aren't counting towards the 1 exotic decryption goal.
This is also happening to my husband and me. We just got back on Xboxes and started playing Destiny 2 after Forsaken launch. We spent a lot of time playing original Destiny, though. We are unable to complete this quest at all. Extremely frustrating and it seems not to be on Bungie’s radar at all.