For technical reasons that are out of my depth (not a programmer!), a given video can only be added to a single item in the database. Tom has chosen to add that video to the Publisher page, so it cannot also be added to the Game page.
It's a technical reason that I'm sure could be very easily changed, but I'm hoping the development team is currently 100% working on getting the necessary changes in place so the redesign can be pushed to the rest of the site
While not ideal, you could always create a forum thread in the game that links to the BGG page for the video.