The controller does not work on the TV because it does not have Bluetooth support. Do you have a physical Steamlink?

If so, it can work, but it's a bit complicated. You have to start the Steamlink with a wired controller, then switch the wireless controller to pairing mode and when it appears on the display, unplug the wired controller.

Unfortunately, you have to do this every time you start the game if the controller has been connected to another device in the meantime.

Give it a try and let me know if it worked.