How do I choose select content not to show on my devices/profile

… seeing and/or hearing certain images and/or sounds as previews on our Apple TVs.

1. You can prevent preview videos from playing automatically. Reduce screen motion on Apple TV - Apple Support
2. You can prevent certain automatic “background” preview videos from playing with sound for the Apple TV app. Adjust Apple TV app settings - Apple Support
3. The ‘Top Shelf’ (space above the content tiles) on the Home Screen will show content related to the highlighted app in the first row. Content from any app wouldn’t show there if the app tile was moved to a lower row. Or, if you group any amount of apps in an app folder (on the first row), then those apps will show generic static images instead of videos. Customize the Apple TV Home Screen - Apple Support
4. For the Top Shelf with Apple TV app related content, you can choose between recommendations (What to Watch) and your Up Next queue list. The former may play unwanted videos, the latter wouldn’t do so. Adjust Apple TV app settings - Apple Support
5. For the Top Shelf with Apple Music app related content, you can choose between Library+Top Albums and music video playlists. The former wouldn’t play unwanted videos, the latter may do so. Adjust music settings on Apple TV - Apple Support

It is currently not possible to prevent select titles from showing altogether (other than age rating by active account), neither from manual selection, nor from already being watched.

For suggesting different behavior, please give feedback to Apple directly: Feedback - Apple TV - Apple.