apt-get install lightdm
And it worked even before a reboot. After a reboot, I had a different login screen but the actual desktop looked the same. The .Xauthority file is now 51 bytes big and I suddenly have a .xsession-errors, which contains a list of environment settings taking place... However, the machine then silently crashed that night and again the following morning. There was nothing about the crash that I could couldn't find a specific cause in the logsbut there did seem to be a number X and GNOME problems: journalctl -b -1 journalctl -e-since "1 hour ago"
I ran an update from the GUI, which may have helped. However, there was a warning about an issue with a screensaver the first time that I loaded lightdm, and the crashes seemed to happen sometime after a clean boot. So I uninstalled lightdm, and installed gdm (which failed as installed already) and rebooted but got no GUI. Then I uninstalled and reinstalled gdm and everything seems fine now.