Apple TV supports select bluetooth wireless game controllers. I don’t recall Nintendo Switch Joy being listed among them. Are they listed as MFi (made for i-devices)? Does Nintendo say that these should be compatible? Compatible controllers will have ‘driver’-equivalent software embedded in tvOS/iOS/iPadOS/macOS/VisionOS. That excludes controllers that don’t have that necessary piece of software for them embedded.
Connect a wireless game controller to your Apple device - Apple Support
Connect a PlayStation wireless game controller to your Apple device - Apple Support
Connect an Xbox wireless game controller to your Apple device - Apple Support
Notable MFi controller has been for a long time: SteelSeries Nimbus+.
Supported and paired controllers should already work outside of the game app, showing almost all of the remote functionality: Up, down, left, right, select, back.