Stadia’s controllers were custom-made to connect directly to the Internet, reducing lag and allowing for instant firmware updates and (sometimes painful) connections to smart TVs. There’s Bluetooth inside the Stadia controller, but it’s only used when you’re setting up Stadia, either with a TV, a computer with the Chrome browser, or a Chromecast Ultra.
I think Stadia was a good idea in concept but terrible in execution. GeForce Now makes a ton more sense from an ownership model. That said, it’s terrible that these controllers can’t connect over Bluetooth. Who knows, maybe these will be super cheap on the used market and still usable because you can still use it wired.