From what I've read I gather that it is now possible to do what I'm trying to do, but:
Is there a way to authorize Moodle (or any web app) to show private Youtube/Google Drive videos?
I know about the "unlisted" option, but the link can be passed around. Downloading from Google Drive to Moodle fills up the small server that we have + we lose the nice built-in viewer that can handle for example .mov's with ease.