The Stream Tab in Teams is a weak integration. Storage and updating has to be done in SharePoint or Teams. Playing the videos directly from SharePoint or Teams doesn't work (there is button in Teams, but it is only designed for small videos). For the team to be available to view the videos they have to download them or you have to manually update the video in Stream every time you have the updated the video in your SharePoint site or Team.
A "publish to Stream" button and "view video in stream" functionality in Teams and SharePoint would remedy much of this.