I think the Xbox requires some kind of authentication chip for USB audio devices, which is why there are separate Xbox and Playstation models of many gaming headsets. You'll wanna get one of the gaming-oriented DACs mentioned here that specifically have Xbox support.

The other option is optical out from either TV or HDMI splitter, and then connecting just the microphone to the controller. But apparently that can have issues with in-game voice chat being routed to the controller instead of the HDMI output, not sure if there's a workaround or fix since I don't have an Xbox.