I'd be surprised if they didn't support the controller, since it is a core part of the Stadia API. Some PC players also like controllers, so I would expect support.

As I understand it, the Stadia hardware is pretty standard CPUs, but GPUs without output logic/hardware. The game's screen images are output to memory, and compressed to a stream to be sent over the Internet to the Stadia Client, which decompresses and displays the images.