Trying to solve another issue, I unistalled and reinstalled lightdm. After that, lightdm failed to start, and I could see this message in journalctl -xeu lightdm.service:
Failed to load configuration from /etc/lightdm/lightdm.conf: Key file does not start with a group
The content of /etc/lightdm/lightdm.conf BTW was
greeter-show-manual-login=true
I got lightdm to work by adding [Seat:*] at the beginning of the file:
[Seat:*]
greeter-show-manual-login=true
IIUC, the problem happened because this file is supposed to be an INI file. Apparently this format needs any configuration inside a group. [Seat:*] is a kind of default group for lightdm, so adding it seemed like the best option, and it worked.
I have no idea of what caused this file to be this way, though.
(Asking and answering my own question in the spirit of Can I answer my own question? and because that message appears ALL THE TIME when misconfiguring lightdm across the years and NOBODY ever explained it.)