Maybe I am not using the precise "embed" url which might be different from the "page" url.

Similar login video issues have been noted on the wordpress platform.

https://wordpress.org/support/topic/unable-to-see-embeded-youtube-videos-as-logged-in-admin/

https://wordpress.stackexchange.com/questions/213732/iframe-with-youtube-video-appears-for-logged-in-users-but-not-for-incognito-use