Jaksta Media Recorder has stopped downloading from Youtube. Putting the video address in the Jaksta window does start looking, but apparently Jaksta can't find it and stops. No download. I don't actually think it's about BigSur, as it is not so long ago that I could download video's from Youtube. It can be a system update from my MacBook Pro, or Youtube has changed its protocols to prevent these downloads.
It's very disappointing.
Have you tried doing a basic reinstall?: uninstall (delete from Applications), reboot, reinstall?
I have done some testing with YouTube videos and it's working for me. We have also not had any other reports of YouTube captures failing.