Because extensions are not part of the video. They are on top of the live video and are meant to enhance it, not for the replays.
oops! is there a way to do recording with overlay?
Not really no.
The alternative would be to run this not as an extension but as a browser source and the stream would include the browser source in their streaming software and then it would be recorded to video.
If the original intent was to record it to the stream then an extension was the wrong technology selection from the get go