Not being able to get the game to show up on 3rd party recording software just seems strange to me when I'm logged into my own account, same kind of problem happens when signed onto other PCs as well.
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You are not banned nor restricted. both of those would give you a message advising of such. Game recording/screen capture software may work differently with Destiny 2 than other games as D2 does not allow any sort of injection which includes game capture. for more info see here: https://help.bungie.net/hc/en-us/articles/360049199891