Hello

Update the firmware on the Apple TV 4K or Apple TV HD, and update iOS on your iPhone, and try again.

Ensure all devices are on the same Wi-Fi network.

If you have screen time enabled, disable it and try again.

Also see ➡️ If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support

If you try this again without the YouTube app involved, such as AirPlay from a YouTube video from within Safari, or from some other local not-the-YouTube app, does that work? (Is this specific to one app, or to all apps? If limited to one app, check with the app support folks for that app.)

Last step in that previous link (above) is this one: If you still can’t use AirPlay after trying these steps, contact Apple Support.