I can't use my photo stream in the system Big Sur

There are two possible reasons, why My Photo Stream might not be available.

• If you Apple ID has been created only recently (not older than three years), you may not be able to use My Photo Stream at all. It is only availably for users who have used it already before Apple stopped the access to My Photo Stream for new users. My Photo Stream - Apple Support "If you recently created your Apple ID, My Photo Stream might not be available. If My Photo Stream isn't available, use iCloud Photos to keep your photos and videos in iCloud."
• My Photo Stream cannot be activated, if you have iCloud Photos enabled. My Photo Stream is a very old legacy service that would be causing complications, if you enable it together with iCloud Photos on a Mac.

Does any of this apply in your case? Is iCloud Photo enabled in the Photos Preferences > iCloud? Or are you using a relatively new AppleID and have not yet used My Photo Stream previously?