I guess you are trying to download and install Big Sur 11.7.10, right?

Does it work via Safari from App Store when booted to safe mode?

How to download and install macOS - Apple Support

Do you see it in Terminal list and can you download it via Terminal:

softwareupdate --list-full-installers

softwareupdate --fetch-full-installer --full-installer-version 11.7.10

Can you access it or some older version by booting to Recovery or Internet Recovery:

Command-R

Option-Command-R

Shift-Option-Command-R