Is it possible to create an extension that will load a video from a third party site and then show the video to viewers on the corner of the broadcaster’s screen?
Technically it is possible as long as you don’t use any embedding/iframe to display the video, it would have to be a video element.
To comply with the guidelines you’d also need to ensure there is broadcaster control of the content, so you couldn’t just have viewers submit videos without them first being reviewed by the broadcaster/mods. If the videos you want to show are you own content then that shouldn’t be too much of a problem, but if you want to allow broadcasters to distribute arbitrary videos through your extension, which they may or may not own the rights to, then it may fall into bit of a grey area and not be allowed.