That would be quite cool. I don't know how to do it, or if it is even possible, but if you come across some resources or anything I'd be keen to check them out. I assume it is possible since Chrome can do it, but I don't know if that has extra / special permissions to do so.

Hi, from the readme i can see you use ffmpeg to stream unknown video files. ffmpeg is able to capture the desktop or only contents from single windows. here its describet: https://stackoverflow.com/a/31076844 and here ist the ffmpeg doc about capturing https://trac.ffmpeg.org/wiki/Capture/Desktop
Is that a way this could work or do i misunderstood the use of ffmpeg in this project?
If i could help you with more research, please let me know.

I also found this: home-assistant-libs/pychromecast#234

I didn't realise ffmpeg could do that! You're correct, we should be able to use ffmpeg to cast the desktop since we already use it for transcoding. When I find some time, I will look into trying this out, it might be simpler than I thought :)