It seems odd at first — the Stadia Controller will use wi-fi to connect for gaming. Why would they do that? The answer is simple: latency.
Google’s biggest hurdle isn’t necessarily in raw internet speed (though that’s important), but it latency — the amount of time it takes for you to press a button, send that info to Google, have them simulate the button press in a game, then send that video to you. It needs to all happen in way under one second, so every little bit helps.
Bluetooth is plenty fast, but it introduces an extra step. You press a button, it sends that info via Bluetooth to your computer/phone, who then send that info out via wifi to Google.
With a wi-fi enabled controller, that button press goes directly out via wifi, saving the bluetooth step and shaving precious milliseconds from the process.