Dolphin is a popular mobile and desktop emulator for GameCube games, and has had support for Android TV for a while. Version 4 had this handy feature that allowed you to customize the key bindings. My NVIDIA Shield gamepad is not the same as GameCube controller, and I am able to point certain buttons to equivalent actions in the GameCube games. With Dolphin 5, this feature was temporarily removed with a revamp of the user interface. This meant new users would need to downgrade to version 4, update the controls, then upgrade to the most recent version.
In the most recent versions, I tried version 5.0-2178, there is a new menu icon at the top shaped like a game controller. Clicking this allows you to set up key bindings for four unique gamepads. This makes it easy to switch a few controls and resume playing without any more changes.
Dolphin has had support for different graphic types, including Vulkan, although I have found it was a little buggy on the Shield. This is evidenced in the recording above, showing the game crashing halfway through. Instead one can use OpenGL, which may have worse performance but greater stability.
You can download Dolphin for free online, although you will need to separately download ROMs.