There really isn't any Lightroom CC now. There is Lightroom Classic and there is Lightroom. We prefer not to hear terms like "up-to-date", but would rather hear exact version numbers. So, what is the exact version number you are using, and what operating system are you using?
I'm a Windows user who knows nothing whatsoever about the Mac. Please describe in as much detail as possible what happens when you attempt to export images. Include any error messages that appear. It's probable that you will have to wait for a reply from someone else who is familiar with the Mac and the Big Sur operating system. But the more information you can provide the more likely it will be that you will receive some useful assistance.