I have both the Nvidia Shield TV along with a Steam Link, and while the Shield utilizes Gamestream which does make the stream clearer than the Steam Link, I just have had never ending problems with it.
When I play a Steam game such as Overcooked, which is a couch co-op game, pressing any button at the start to activate the controller as player 1 will automatically put two players into the game instead of one, as if it double presses any button I press. This makes the game unplayable. In addition to this, when I add a PS4 controller to my Shield, it will automatically take control as player 1 in any game I try. Then, when I try to join a party in a game like Rocket League and play split screen, it'll put both player 2 and player 3 in the game, even though there's only two controllers connected.
Why is the Nvidia Controller double pressing in Gamestream? It works fine in Play Store games and navigating the menu, but games that work perfectly fine on the Steam Link just seem to struggle on the Shield.
If you're launching games from Steam, make sure you enable Multiple Controllers in Steam and then AGAIN when the game launches. I find it has less problems if you launch directly from the Gamestream app