the Apple TV Gen 3 is stuck on data & privacy screen.

Please use the button combination procedure instead.

1. Unlinking any previous remote: press and hold the Menu and Left buttons for up to six seconds. On screen, look for broken link symbol above an icon of a remote.
2. Linking the (new) remote: press and hold the Menu and Right buttons for up to six seconds. On screen, look for a linked chain symbol above an icon of a remote.

Is this about the older HD 3rd generation model, or about the latest 4K 3rd generation model?

Identify your Apple TV model - Apple Support

Find your Apple TV model number - Apple Support

Possibly the linking or pairing between the box and the remote is lost, and needs to be re-established.

• For Apple TV (4th generation or later) and the Siri Remote, see: If your Siri Remote or Apple TV Remote isn't working - Apple Support
• For Apple TV (3rd generation or earlier) and the old style Apple Remote (aluminum), see: Link or unlink your Apple Remote (aluminum or white) from your Apple TV - Apple Support - Wayback Machine

Does the status light blink on the box when pressing any of the buttons? That would mean that it at least detects a signal.

Try exchanging the battery for a new one. How to replace the battery in your Apple Remote - Apple Support.