when searching for mini mac software update gear keeps spinning and nothing else happens.

Hello,

This article is primarily about errors after an update has started but it could also be helpful for starting an update. It's not mentioned but a VPN or 3rd-party firewall/content blockers could keep your Mac from connecting to the update. Using Safe Mode should help resolve this if you do not want to or cannot uninstall them.

If an error occurred while updating or installing macOS - Apple Support

"Install in safe mode

Install while your Mac is started up in safe mode. To start up in safe mode, determine whether you're using a Mac with Apple silicon, then follow the appropriate steps:

Apple silicon

1. Shut down your Mac.
2. Turn on your Mac and continue to press and hold the power button until you see the startup options window.
3. Select your startup disk, then press and hold the Shift key while clicking “Continue in Safe Mode.”
4. Log in to your Mac. You might be asked to log in again.

Intel processor

1. Turn on or restart your Mac, then immediately press and hold the Shift key as your Mac starts up.
2. Release the key when you see the login window, then log in to your Mac. 
3. You might be asked to log in again. On either the first or second login window, you should see ”Safe Boot” in the upper-right corner of the window."

I hope this helps.