I get a constantly spinning wheel saying "checking for Updates". I doesn't resolve to say an update is available or the Mac is up to date

Yup - the Apple Software Update Applet has and still has issues for updates . Not only for Big Sur but even older versions too.

Suggest restarting in Safe Mode. This will perform a Disk Repair, clear cache files and only load Apple Software, extensions and fonts. The boot up will be slow and can take some time - Normal.

The attempt the Software Update applet again.

If still no success - restart in Normal and they once again.

Gosh - it still not successful restart in Recovery Mode >> Command + r << and Choose Reinstall macOS. This is not a typo or some other error.

The Recovery Mode Reinstall macOS should bring down the latest Version of Big Sur 11.6.4 and commence the installation.

It will not Touch personal files, setting, preferences or installed Applications.

Good computer practices dictate doing a Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS Before attempting any or all of the above