The stadia controller is by far the best controller I have used but, it would be cool to not just have bluetooth but to have an app for pc where you could connect the stadia controller to so that you could play different games on the pc without a wired connection. The only way i've gotten the controllers to work on steam is by following this reddit post.
To play other non-steam games with the stadia controller you have to add the games to your steam library by clicking the checkmark on the bottom of your game library, select non-steam game, and checkmark the games you want to add. once the non-steam games are in you steam library and you've followed the reddit post to get your controller working with steam, then launch the games through steam and it will work. Thats how you get the stadia contoller to work wired but there is no way to play wirelessly. Stadia needs to make someway to play wirelessly by some type of update or software.