The Steam Controller cannot be used on Steam. Destiny 2 is specifically targeting the Steam Controller and we can't have Xbox Glyphs and/or core functionality, any input just doesn't get registered.
I'm using it wired.
The Xbox 360 Gamepad works, and I guess DualShock works too.
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Yeah probably not supported yet. Xbox One Bluetooth and 360 work fine, since that's connected through Bluetooth via PC.