If there's one thing that Stadia needs in regards to its Android use, it's support for playing games with the Stadia controller in a wireless state. That may be changing soon though as it looks like Google is preparing to add it in the near future.
Currently you can only play games on your Android device if you have the Stadia controller connected with a USB C cable. This isn't a terrible experience but there are a few issues with it that could pop up from time to time.
The main issue being the lack of ability to play and charge your phone simultaneously. Except for perhaps in very special cases. For example, if you have the ASUS ROG Phone II you can easily plug the controller into the bottom USB C port and use the one on the side for plugging the phone into the charger.
Most users don't have this option though. So they have to make the choice of playing or charging.
Wireless Stadia controller support on Android suggested by new app version
According to 9To5Google, the latest version of the Stadia app points to the wireless support on Android being made available soon.
Strings of code within the app's APK mention the ability to link a controller to the phone with the linking code. If you play Stadia games on the Chromecast Ultra, you'll know that the linking code is how you connect the controller to your CCU and play games on the TV or monitor it's plugged into.
The linking code is just a series of buttons you press in a specific sequence to get the two devices to pair up. It works with little to no issues on the Chromecast Ultra. So hopefully it'll have the same success when it comes to Android.
You can still play wirelessly now with a different controller
If you can't wait for this feature to pop up properly with the Stadia controller you have other options.
You could always use a different controller. The DualShock 4 for the PlayStation 4 is one great example. This supports a connection to the Android device through Bluetooth. So you can easily connect it and begin playing.
You could use other wireless controllers as well. Though the DualShock 4 might be your best option. Or at least one of the best because of how well-made it is.
That being said, the Stadia controller was specifically designed for use with Stadia. So features like the Google Assistant button and the Capture button are there to enhance the experience. Which you'll need the controller for.