After searching endlessly for a fix to this issue, I have resorted to posting what I am experiencing.
I handle LogMeIn on (let's just call it) a lot of Mac computers for our organization.
I have repetitively run into an issue with Big Sur & Monterey when remoting into a client computer that looks like the following (logos/client's name have been removed):

When this happens, I cannot see anything on the client's computer, however, my mouse movements & clicks WILL register.
The only known fix is to physically visit the client's computer and un-check/re-check all security settings for LMIGUIAgent. This poses an issue as some of our client computers are sent worldwide after preparation.
I did see a post recommending trying a caffeinate command for Macs with the M1 chip - our Macs have Intel chips.
I did see a post recommending lowering BIOS security levels - we can't do this. We must have Full Security & a BIOS password set for every machine.
Sample Computer (experiencing issue) Specs:
- Mac OS Monterey (12.2.1)
- Mac Mini (2018)
- Intel Core i5 3 GHz 6-Core Processor
- 8GB RAM
- Intel Graphics
My Computers Specs:
- Mac OS Monterey (12.2.1)
- MacBook Pro (13-inch, 2020)
- Intel Core i5 Quad-Core Processor
- 16GB RAM
- Intel Iris graphics
Any input on this issue would be appreciated because it has truly become an issue within our organization.