If you streamed Stadia to your web browser, sure, you could use an Xbox controller. The real feature of the Stadia controller was that it connected directly to the Stadia servers to minimize latency (and load on your computer), instead of connecting to your computer using USB or Bluetooth and having your browser relay inputs. You’d initially set the controller up using your phone and Bluetooth to bootstrap a Wi-Fi connection, and from then on it would talk to the servers over Wi-Fi and TCP/IP.
As far as I understand it, Amazon Luna’s official controller does the same thing.
That doesn’t seem like a good idea to me. You reduce latency by removing Wi-Fi entirely. The latency between the controller and a machine sending to Google is comparatively minor. And that machine can have a wired network.
AFAIK that means that all of those Stadia controllers are ewaste now unless Google releases a firmware update that allows them to be paired as Bluetooth.
You can use them as a wired controller with the existing firmware (although I agree it would be ideal if Google released an update adding bluetooth functionality as well).
As far as I understand it, Amazon Luna’s official controller does the same thing.