I have had the issue where the controller will connect but no button / axis events actually are produced. Disconnecting and reconnecting fixes it for me. This seems to be a common issue from other comments I've seen online.
And the Stadia controller was very slightly laggy even when just using the USB cable.
I've been having some annoyances which seem to stem from somehow any BT use in this laptop causing it to lag sometimes like it's competing in the latency olympics, and strange interactions like it not allowing me to pair any bluetooth game controller if I have a wireless headset paired (it just errors out every time), but the reverse not being true, so I can pair them if I do it in the right order, but have to unpair both if the game controller disconnects for some reason. :/