My brother and I have been playing Knights on our TV (through Steam Link) with Steam Controllers. It's fun to play with controllers instead of the keyboard. The only glitches seem to be navigating the menus with the controller is clunky, and Knights stops working if you bring up the Steam Big Picture overlay while Knights is in the menu screens.
You have to select "Add a Non-Steam Game to my Library" to launch it for Steam Link, and you have to deselect "Allow Desktop Configuration in Launcher" to have the controller bindings work. I'll post my Steam Controller bindings file here once I get it perfected.
Thank you for sharing this information about playing Knights on TV with Steam link, very fascinating.
I'm researching that system and controllers you mentioned, looking forward to more of your settings details if you figure out more, and I wonder if there's anything Stephen could do to make a version of Knights that resolves any of the menu issues you mentioned when playing specifically through a steam link.