If I disable YouTube Integration Essentials, no YouTube videos are shown at my site!

This is expected behavior. Deactivating the add-on (which is displaying the YouTube videos) will disable them completely (Note: They are still in your messages, they are just not displayed).

If you do not like to continue to use our add-on, please uninstall it.

If you uninstall it, the YouTube videos will be shown by the default XenForo YouTube media BB code handler again.