Turns out, that Stadia app update earlier today, which expanded support to more Android devices, has a small unintended side-effect that Android TV users can take advantage of. If you've got a mouse and a Bluetooth controller, you can sideload the app to get Stadia working on your Android TV.
According to our friends at 9to5Google, it's a surprisingly simple (if slightly awkward) process. Just sideload this latest version of the Stadia app — which is to say, download and install it manually via our sister site APK Mirror — fire it up via the Android TV launcher, and use that mouse to follow the setup process, since the Android version of the app doesn't play nice with Android TV's controls. According to the video above, it's about that simple.
Once you're logged in, it looks like you still might need that mouse to get past a new prompt — Google considers support for all but a handful of phones as "experimental" right now — and you'll also be stuck in a weird portrait mode until you launch a game. Still, it works! Also note, @androidtv_rumor says that the Shield remote app circumvents the need for a mouse.
Guys, Stadia works on Android TV !
Sideload the app, go into settings and enable the option for any device in "Experiments".
Plug the controller to your Shield for example and use a mouse or the Shield remote app to accept the prompts. Enjoy !#AndroidTV
— Android TV Guide (@androidtv_rumor) June 11, 2020
9to5Google tested it with an Xbox controller and Google's ADT-3 Android TV developer device, but it should probably work with other hardware as well. The hardest part is sideloading the app. On Android TV, that's a bit of a process (you'll need to allow apps from unknown sources in Settings and then find a way to access the APK either through cloud storage or some other method) but if you can get it working, you can enjoy Stadia from your Android TV.