Thank you for your reply, GarryP. I was about to attempt a full-screen screencast, as you suggested, but then I decided to try one more thing. I quit Designer and then restarted it, then opened a sample Designer file from the opening splash screen. The layers worked fine on that file. So I then opened a new blank file, added a few objects on different layers, and tried manipulating those objects and layers and, lo and behold, everything is working fine. The strange thing is I had quit Designer and tried to open or create new files multiple times before but the behaviour had not resolved. And yet for some reason, the glitch seems to have disappeared this time (and with other new files since then).
I have no idea what caused the Layers Panel problem before, but at least it's now working.
Thanks again for your attention to this question.