@Nightcrow I play on Chrome with a wireless Xbox 360 controller without any issues. I just have a Microsoft Xbox 360 Wireless Receiver installed and just connect with my old Xbox 360 controller just fine. I still need to have access to a keyboard if I want to access Stadia menus and the like, but once a game is running no issues with an old controller.
@Nightcrow : It's not so much the assumption, just that Google haven't yet released a controller with more universal WiFi capabilities. Microsoft was slow for the Xbox 360 to make the controller work with a PC, and they were slow again with releasing a dongle to allow an Xbox One controller to connect with a PC. Same scenario with Google IMHO. I'm sure that something will get released later this year to fill that void, but until such time, there are possibilities even if it involves a 3rd party controller.