Reference: https://discussions.apple.com/thread/253467325?sortBy=best
After a million tries.... the following process worked! This is using all legitimate tools, so no drama!
To fix failed activation, and therefore restore AirPlay capability, on a Mac, install Apple Configurator 2 from the app store.
Plug in your Apple TV 3rd Gen to power and to a Mac using a micro USB cable from the Apple TV to the Mac.
In the configurator, perform an Apple TV OS "Restore" using the "Prepare" gear icon in the top menu bar (it should prompt you that you must do this anyway).
Once it has been restored using the configurator, do what the prompts say: "Unplug USB cable and power."
Then, "Power it back on" and wait for it to boot.
When the status light is solid (booted), plug in the USB cable connected to your Mac again. You will then see a "mirrored" display of a TV and Apple TV icon with the setup screen as if it were plugged into a TV.
Click on it, and it will then display the Apple TV home screen and version info, etc. on the right. A menu of 4 items will now be available in the left-hand column of the configurator app.
Click on the second item "Apps". You will see a message "Apple TV not activated".
Click the button that says "Activate".
Bam!
Unplug from Mac and power, plug into the TV you would like it on, and carry on with a normal, activated setup!
Hope this helps - let Apple know that they should use this method. IT WORKS!