"Whether I start System Preferences in the Applications folder OR select System Preferences in the Apple menu, I see the same screen. The icon on the top left will not stay set! I can select it, change it, and as soon as I quit the System Preferences and re-start it, the icon reverts to this drum! This drum is NOT even available to select in the first place!!"
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Thank you for the screenshot.
Troubleshooting Application Not Working as they Should:
A. Single-Out this User::
For troubleshooting purposes, create a new administrator user, and see if you still get the issue. If so, then it is an issue with the entire macOS. If not, then it is an issue pertaining to this user.
B. Reinstall the macOS:
Also, one option would be to reinstall the macOS to fix any bugs in this app. Nothing would be lost. It's just that misplaced items would be placed back to where they should be and corrupt system files would be replaced by clean files. Go Here: Reinstall macOS - Apple Support. Boot into Recovery Mode (Intel or Silicon). Important: Prior to installing an update or performing a reinstallation, to prevent data loss, be sure to Back up your Mac with Time Machine.