Connect a wireless game controller to your Apple device - Apple Support
DualShock 4 is listed as a compatible bluetooth device, when your Apple TV system software is reasonably up-to-date (tvOS 13 or later).
Failed to pairing …
… it took forever to detect my Dualshock4 controller.
But it did eventually? Once paired, the Apple TV should remember the device, and be able to connect much faster.
FAQ: How to use DualShock 4 with your Apple devices - Playstation.com
Q. How do I pair my DualShock 4 wireless controller with my Apple device?
A. If you are using a controller with your Apple device for the first time, you will need to follow the steps below to pair them together using Bluetooth:
- Make sure the light bar on the controller is off. If the light bar is on, press the PS button until it turns off. If a USB cable is connected to the controller, disconnect it.
- While pressing and holding the Share button, press and hold the PS button until the light bar flashes.
- Enable Bluetooth on your device, and then select the controller from the list of Bluetooth devices. When pairing is complete, the light bar turns a solid color.
and
Use Bluetooth accessories with your Apple TV - Apple Support