Watching videos and playing games on your Chromebook usually uses a lot of power. This can make your computer overheat and restart. Perhaps this only happens in Administrator Mode because Guest Mode doesn’t have as many rights to your computer’s resources (like downloading, etc.) and therefore uses less power?
Since charging your Chromebook can also produce heat, try playing the games without it plugged in and charging. If it still restarts, you may have to continue doing what you have been doing do far - downloading the media content as an administrator and then playing the games in Guest Mode.