@JoYo That's definitely not by design. Just to make sure, you own Shadowkeep outside Stadia, yes? Also, which quest is this? Is there a chance the quest already expired? If you can provide screenshot that would be great too.
I do not own Shadowkeep outside of Stadia.
Bungie gives access to expansion and season pass content across all platforms.
For example, I bought Forsaken on PC and can play the content on PS4.
The quest is the main quest for Shadowkeep that doesn't expire.
Direct links to screenshots.
OK, apparently I was partially mistaken.
Only the season pass and other digital content are transferred with cross-save.
The expansion packs are not.
I will need to buy Shadowkeep expansion on Steam to play Shadowkeep content when not playing on Stadia.
@JoYo Yes you will need to purchase Shadowkeep on every platform you want to play on. But let's say you play Shadowkeep on Stadia, and got weapons/armors exclusive to Shadowkeep, you will still be able to use them when you play it on platform without Shadowkeep. And this rule apply to any expansions not included in the free version of Destiny 2
I have the same problem but I actually bought Shadow keep on PS4 and when I linked it with stadia it also had Shadowkeep but when I came back to PS4 I didn't had Shadowkeep anymore and I had to buy it back, like if stadia stole the dlc