This is a lie. Your Stadia controller is using WiFi to connect. The app absolutely does not need to know your GPS coordinates to find your controller. Google are blatantly lying about this to trick you in to granting them access to your geographic location. This is probably illegal and if it isn't it should be. Spread the word.
No you are missing the point. This is not a requirement to get a device to join your WiFi network. At worst this is incompetent programming and let's not kid ourselves here. This is deliberate. Google home does not require location access to set it up to join my WiFi network but Google Stadia does all of a sudden?
Were you able to set up your controller with your phone's Bluetooth turned off? Because I sure wasn't.
This suggests to me that it's actually using Bluetooth to detect nearby devices (which requires the location permission) in order to discover the controller and then tell it which WiFi network to join and what credentials are needed. That's a pretty standard approach.
@AreWeThereYet Do you have proof that Google is lying? Trying to be devil's advocate here, just because the controller is using WiFi, that doesn't mean that it's enabled to find nearby controllers. It might be just the matter of design choice, that they want to use location access instad. At the very least, they were being honest and upfront about using location access (which you can disable) to find nearby controllers.
The location access is a quirk of the security system on Android related to using Bluetooth. It isn't actually using your location for anything, though I can understand how this isn't obvious. Almost all apps on Android that utilize bluetooth for some reason will ask for the same permission.
So don't worry, it's not tracking your location it just wants to use Bluetooth to set up your controller.