I too am facing this issue. In my case, I fully rebuilt my iMac a couple of weeks ago - to overcome the growing performance issues I have seen since the Big Sur releases. I'm on the latest of everything. So, I guess that means I get all the new and untested functionality Apple has to offer.

Brand new Mac mini (M1, 2020) running MacOS Monterey Version 12.2.1. And yes, Screen Sharing is broken and will not enter Full Screen. It's certainly not the only problem on this OS. You cannot upgrade your way out of Monterey flakiness. :(