Option To Treat Videos As Live Feed
EasyWorship already does this. If your video backgrounds are starting over when you go live to a new scripture, then you don't have it set as a background. You have it set as a foreground element. Make sure when you create your theme that you're setting it as a background. You can go to Inspector > Background and select it there. Or, you can click the Media button on the toolbar in the theme editor and drag and drop the video you want to the thumbnail on the left side of the editor.
For instance, if a video is selected as a background for scriptures, if another scripture is sent to the live output, the video restarts, usually on a whim.
An option, perhaps on the theme level, can be used to control if a normal background video behaves like a live feed or not.