When I initially tried researching this topic, the information I found was a couple of months old, and leaned towards the button mapping being messed up.

I finally got one of the controllers and tested it. It worked great with emulators, but that was kinda expected due to the ability to remap buttons. I also tested Doom (2016) and Titanfall 2 via gamestream, and both controlled as they should have.

You may have to update the firmware on the controller prior to using it on the shield TV. You can update it via an Xbox one or by the Xbox accessories app on Windows (10?).

I hope this helps some people trying to look for a definitive answer on compatibility.