Hello and welcome to Apple Support Communities, Chelecool!
We understand you're having trouble pasting into Apple TV while using iOS 16. We'd be happy to provide information to help you address this.
Try the steps here: Set up the Apple TV Remote on your iPhone, iPad, or iPod touch - Apple Support
"If you can't set up the Apple TV Remote
Try these steps. After each step, see if you still need help.
- Check that your iPhone, iPad, or iPod touch is on the same Wi-Fi network as your Apple TV or AirPlay 2-compatible smart TV. On your iOS or iPadOS device, go to Settings > Wi-Fi. On your Apple TV, go to Settings > Network.
- Make sure that your iPhone, iPad, or iPod touch is updated to the latest version of iOS or iPadOS, and your Apple TV is updated to the latest version of tvOS.
- Restart your Apple TV or AirPlay 2-compatible smart TV. Also try restarting your iPhone, iPad, or iPod touch.
- If you still can't set up the Apple TV Remote, contact Apple Support."
If you continue having issues after trying those steps, reach out to Apple Support for assistance:
Contact Apple for support and service - Apple Support
Have a great day!