spinning update circle

No Updates or No UpGrades ?

The two are not the same.

Updates would be Big Sur 11.6.5 to Big Sur 11.7.6

UpGrade would be Big Sur to the a newer version of macOS this computer Qualifies to run.

Does this computer appear as Listed and qualifying for an upGrade to a newer version of macOS? 

Computer Qualifying for macOS 13 Ventura 

Computer Qualify for the Monterey upgrade

The Age of this computer will Govern what newer version of macOS it will Qualify to install.

Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install. 

I understand Software Update is "Checking for updates..." with no end in sight.

Several workarounds have been proposed, such as restarting your Mac in "Safe Mode" and checking for updates while in that mode, and / or resetting NVRAM. Try both.

How to use safe mode on your Mac - Apple Support

Reset NVRAM on your Mac - Apple Support