For those of you who really want it.
Now that Stadia has rolled out its free tier, Google is continuing work on bringing more much-desired features to the platform. This week that feature is the ability to wirelessly connect your controller to your PC and play without the cable.
Starting this week, you can now use the Stadia Controller to play wirelessly on your laptop and desktop. You no longer need to connect your Stadia Controller physically to your computer to play your favorite games. You’ll be able to link your Stadia Controller and Stadia.com through your Wi-Fi network and play without a USB-C cable connection.
It should be noted that the Stadia controller does not connect to the PC but rather connects to wifi and is then used to play on a PC on the same network. You aren’t connecting the controller via bluetooth so it won’t work with non-Stadia games.
Linking your controller is simple; you just head over to Stadia.com and click on the controller button in the top right and follow the standard sync procedure. Press the Stadia button on your controller and enter the simple button combination.

Source: Stadia