If I want to play some Destiny 2 on Stadia, I turn on the TV with the Chromecast Ultra connected, select the correct input (it’ll be showing the “screen saver” pictures that change every so often) and then press and hold the button the controller until it turns on. Since the controller is paired with that Chromecast, it causes Stadia to launch, as expected. The only potential difference with others is that I have a PIN on my account, so I then have to select my account and enter the PIN. Stadia game selection UI then launches.
What is weird is that shortly after the game chooser launches, the controller will go into a weird mode with the white LED flashing and act like it’s not paired. The only fix for this I have found is to press and hold the controller button until it powers off, and then press and hold it again until it powers back on. Everything seems to work fine from that point forward… but this is my experience with EVERY launch of Stadia. I wonder if I am the only one.