Both Microsoft and Sony wanted Destiny as an exclusive
This is according to former Bungie composer Martin O'Donnell.
After delivering Halo: Reach, Bungie wasn't a Microsoft first-party developer anymore, and their first project was Destiny. As you know, Activision published it, but it could have been very different.
During the GamesBeat Summit 2021 online event, the former Bungie composer Martin O'Donnell (mainly known for the epic Halo soundtracks from Bungie's games) had a couple of nuggets to share. He says both Microsoft and Sony wanted to make an exclusive deal for the game, and it was really close that it ended up with one of them.
But in the end, they wanted to own the Destiny IP, and this was a big no-no described as "non-negotiable." That's how Activision got to publish the game instead, as they simply didn't have this demand.