Is the original remote physically broken, or just not working? In case of not responding, check the battery, and try pairing or linking again. If your Apple TV remote isn't working - Apple Support
What is your model Apple TV? Identify your Apple TV model - Apple Support
The Apple TV (4th generation or later) should be straight-forward to pair with, as described, if both devices are already on the same home network.
The Apple TV (3rd generation) needs a functional remote to set up the Apple TV Remote widget in Control Center.
For the Apple TV (3rd generation), your options are:
The Apple TV (2nd generation) is not compatible with the Apple TV Remote widget in Control Center.
For the Apple TV (2nd generation), your option is: