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.
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.