That’s because Google hasn’t yet brought Stadia to the new-and-improved $50 Chromecast, and its predecessor didn’t support Bluetooth, meaning the only way to pair a controller was to loop through Google’s servers in the cloud.
Thankfully, third-party controller support was quite robust on those other platforms, letting you easily sync a PlayStation or Xbox gamepad to play, and it’s unlikely you’ll even need this new Bridge Mode to play Stadia on the 2020 Chromecast whenever it arrives; the new Chromecast does support Bluetooth game controllers.