You'll be able to use your phone as a Stadia controller for your TV if a feature Google is working on gets a wide release. 9to5Google has decompiled the APK for Stadia update 3.15, which recently started rolling out, and found an experimental feature called "bridge mode" that will give you a way to use your phone's display as a controller when you're playing on TV. Currently, you can only use the service's official controller while playing Stadia on TV via Chromecast Ultra.
The feature doesn't just turn your phone into a touchscreen controller, though. It will also allow you to use a controller connected to your phone to play Stadia on a TV. As 9to5Google says, it's not entirely clear if that non-Stadia controller has to be connected to your phone via USB or if Bluetooth models will also do. A USB connection will likely work better, though, seeing as your phone will serve as some sort of middleman, and a Bluetooth connection could cause additional input lag. To be able to use your phone to play games on TV, the devices have to be connected to the same WiFi network. As with the Stadia controller, you'll have to key in a four-button linking code to pair your devices.
In addition to bridge mode, 9to5Google also found evidence that Google is working on contacts integration for the service. It'll allow anyone who has your phone number or email address saved to their Google Contacts list to add you as a friend. Finally, Google seems to be working on an accessibility feature that'll let you easily toggle Stadia's screen reader on or off by pressing the start and up buttons on your controller.