Google has announced that wireless support for their controller is coming to Stadia this week. Specifically, computers will support the feature. Uniquely, the Stadia controller connects directly to the network via Wi-Fi, not to your computer via Bluetooth, so Bluetooth capabilities are not required. Look for the controller icon at the top of Stadia.com for wireless support. Expect it this week.
A USB C cable is still required to play on phones on the app. However, you can play wirelessly in the chrome browser, at Stadia.com in desktop mode.