Menu bar disappears and all Finder-related tasks greyed out and unclickable after Monterey update

Pulled from Menu bar disappeared and all Finder-relat… - Apple Community with some edits...

• Using Monterey 12.6
• Happening on 3 separate 2018 Mac minis - 3.6 Ghz quad-core intel core i3, 8 GB 2667 MHz DDR4
• Machines are running a "classroom" profile that is set up to auto-login
• Menu bar disappears after a period of time and will not appear on hover over - please note, "Automatically hide and show the menu bar on desktop" and full screen are both NOT CHECKED in the Dock&Menu Bar settings
• Cannot single left-click most items or highlight text, however double-click and right-click works to show menu
• Double-click on Finder, window opens but all Finder shortcuts in the sidebar are greyed out and unclickable. Can right-click them to remove them, etc, but cannot left-click to select/open
• Unable to do anything Finder/system-related, like Spotlight Search
• Cannot type anything on both wired and wireless keyboards
• Keyboard shortcuts do not work
• Dock appears and I can open Launchpad to open apps, such as Terminal, but as you cannot type anything, it is useless
• Can open the Activity Monitor but all the menu items are greyed out
• Cannot take screenshots
• Can control the machine remotely to switch user or log it out and then back in to the same classroom profile, and menu bar and all other functionalities return
• A restart does not always fix the problem, but a forced shut down via the power button on the mac mini does

We have seen only a few posts with these exact symptoms but never a solution - any thoughts on what is causing this, and/or whether Apple is aware of this as a bug? It does appear to be profile-related as signing out and back in fixes it temporarily. The classroom profile is auto-configured via an MDM and there does not appear to be any rhyme or reason why it is happening to only 3 of these machines, out of approximately 30.