Well, I'll be ******. After trying all the various suggestions in this and other threads, I discovered that NOT being in full-screen is what made it possible for me to share my screen in Microsoft Teams. Just the opposite of jack's experience.
There was not an application blocking things for me. If I rebooted and started Teams as the first and only application, I got the error about needing the screen-recording privilege (which was already on). I also checked the firewall enable-incoming option, which was on.
This is a real bear of a problem. I wondered why it suddenly appeared; I thought it was some maintenance that was applied. But no; for me it was switching to full-screen mode.