Ubuntu launching Unity instead of XFCE after gdm3 upgrade (solved)

Yesterday I installed updates on my Ubuntu 24.04 work laptop and rebooted, and was annoyed to find that I was being logged into the Unity desktop, not XFCE. Not only that, but after logging out again I discovered that there wasn't even an option for selecting XFCE. The login screen also looked different. The solution seemed to be to select a different greeter. More details under the cut. I think…