This is more of an FYI for people who have this issue. I have CCU (the one that comes with the Stadia Premier Edition box) connected using Ethernet cable and have noticeable controller input lag in every game. I have great broadband, all connection tests come back excellent, etc. and still have it. This is a problem for games that require precise movements like shooters.
However, there is none when using Stadia wirelessly through Chrome browser on my 6 year old laptop, or Chrome or Safari browser on my iPad Pro. Note: on the laptop you gotta go into Chrome settings and turn 'Use hardware acceleration' off.
There's obviously an issue with the CCU as I've tried everything I can think of to get the input lag to disappear, which is a real shame because it would be nice to play games on the TV screen.
Your TV support game mode? have you tried if in the settings you can enable a game mode? this should reduce any lag on the CCU
Your TV support game mode? have you tried if in the settings you can enable a game mode? this should reduce any lag on the CCU