Maybe the lag is coming from the TV, did you try game mode?Run ,run ,runaround....
Noticed input lag when trying to play games.Have tried A PS4,and Shield controller....input lag still there, maybe a half second. Just enough to ruin the experience. I talked Nvidia, told me it is unusual...to "Go to settings -> Apps -> System apps -> Bluetooth -> Clear cache, data, and force stop the app and restart the Shield
Once the Shield device is back on, pair the remote/controller and check"
Still there after following instructions.....and even after getting a brand new replacement Shield Pro from Best Buy tonight. I guess i wont be playing games on my new Shield. It is a pretty good streamer though.