For what it's worth, the headset is working again. I noticed that there was a Windows update available for .NET and installed it just in case it might help, which forced another reboot. After that second reboot, my headset started working again. The X in the red circle is gone and the dongle LED is lit again. I can hear videos again.
The fact that I needed to reboot TWICE just to make this bizarre problem go away does NOT please me.
By the way, now that iCue can "see" the headset again, I see that my firmware was current at v0.15 and my software was a bit out-of-date at 3.18.77; I'm updating to 3.20.80 as I write this.