Hi. I had the same problem with zoom, especially in sharing my screen and just solved it.
This is not actually a problems from macOS big sur, I think because they updated new term in privacy so you have to unblock it in System Preferences.
Firstly, make sure that you update newest zoom version.
Go to System Preferences -> Security & Privacy -> Screen Recording -> click zoom icon to allow share screen on zoom.
Don't forget to click to the 'block' icon in the left bottom to unblock for making any change in Security & Privacy!
Hope this can help you guys!