I searched for information here and there, tried numerous suggestions, but all without luck. I tried to use the "Install VMWare Tools" command several times to update or reinstall VMWare Tools. This didn't help.
Recently, I have received a notification from VMWare that Tools 11.2 is out. Workstation offered me to download and update Tools. So I did. After downloading, I used the "Install VMWare Tools" menu command to update VMWare Tools. To my surprise, this didn't update Tools! I still had version 11.1.5 installed.
In addition, the machine-in-question didn't let me switch shared folders while the machine was powered on. Workstation told me that I must have Tools installed. That was confusing, but the message also signaled that the host is having problems communicating with the guest. So, I probably need to focus on fixing Tools in the guest.
The next thing I did was visit the download page for VMWare Tools and download version 11.2. There, you can download the "external" package to install from the host and the in-guest installer. I took both but used the in-guest installer.
Running the in-guest installer in the guest machine to upgrade VMWare Tools didn't work again! The files remained old. Then I uninstalled Tools 11.1.5 completely and rebooted the system. On one of the two virtual systems, the Tools' files remained on the disk. So I removed them manually. After that, I used the in-guest installer again to install the "fresh" copy of VMWare Tools 11.2. And voila! VMWare Tools was installed.
As the fresh version of Tools was installed, the Fit Guest Now was re-enabled and appeared to work. The happy end is here (for the time being). I still don't know why Tools got broken - was it some Windows update that damaged them, or the Tools I had were too old for Windows 10 build 2004. Let's see how it goes with the next Windows 10 updates.