I've completed every mission way back in vanilla before The Dark Below expansion and even completed the quests again when The Taken King expansion came out. Yet even so, I don't have access to the white sparrow you get from "The Warmind" mission. I've replayed the mission and never received it for the kiosk.
I deleted the sparrow long before kiosks were even a thing after I worked hard to get the legendary sparrows through the drive upgrades. I don't want to start a character over at this point. I've done every quest except a small few involving the factions, refer-a-friend, and some repeatable ones, and I don't want to have to go through everything again, especially the PvP ones.
Is there any way this can be corrected without having to restart a character? I wouldn't care so much, except for the new Age of Triumphs book. I do know I'm not the only one this is happening to. Several friends have the same issue and they don't want to restart either.
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Unfortunately, if you didn't have this sparrow in your inventory when the kiosks were introduced, it will not appear in the kiosk. The only thing you can do is start another character.