I had a Steam link before the Shield (upgraded when I finally got round to getting a 4K TV) and didn't notice any more lag than the Shield.
But it only outputs to 1080P and no HDR right?
To be able to use the Steam streaming you need to add the executable for the game to Steam as a "Non-Steam Game". This works fine for most 3rd party games (e.g. GOG, Epic, Origin & Uplay), but because of the way the Game Pass games are stuck in a hidden folder which you don't actually have permissions for, you can't do that. I've tried a couple of times with various guides online & tools which supposedly let you do it, but never had any luck
Ah right. Have you tried selecting the shortcut? As this just worked for me via GameStream on GFE. I tried to add the game the normal way but like you couldn't modify the permissions weirdly. But selecting the game shortcut on the desktop allowed me to stream the gamepass games via gamestream to my Sony TV using the Moonlight app