My Chromecast Ultra (from Stadia Premiere Edition) has a combination of these issues. Sometimes my controller stops working in-game, while the game stream still runs flawless and the Stadia button works without lag. I can even navigate without issues in the Stadia in-game menu.
Sometimes my controller input stops working at all, but the Chromecast doesn't recognize that the controller is disconnected and the video freezes too. The controller still has constant white light and my CCU doesn't give the "please pair your controller with XYZ button pressing" pop-up. After restarting the controller, pressing the buttons and waiting for the controller to reconnect, my controller is able to connect again. But after connecting my controller (when it vibrates as confirmation and switches back to constant white light), the CCU throws that pop-out massage and refuses to take input from the controller, that i connected successfully the moment before that pop-up appeared. My CCU is usually really hot when this happens.
After restarting my CCU when I have these issues, my game sometimes starts without video. Working audio is indicating that the game runs and my button presses are working, but I have to restart the CCU device another time, to fix that.
The first issue looks like I have a streaming issue while my CCU and Stadia controller still work.
The second issue looks like the CCU freezes/crashes, because my controller doesn't show me a disconnect, is able to reconnect without issues and the CCU is freezed. The only thing that happens, when everything is freezed, is that the pop-up comes AFTER I've connected the controller. Then it continues not reacting on anything.
How hot are your CCU devices? Maybe it's an issue on my CCU, but it feels really hot when this happens. These crashes happened once a day or once in two days, so it isn't such a great issue to restart m CCU twice, but I'm unaware, where this problem comes from.
The most annoying issue is, that my game sometimes crashes, my controller disconnects and reconnects, but the connect controller pop-up doesn't disappear, even when connecting successfully