Everything OK! Synaptic works again and the backlog of new updates that built up over the past three weeks has been cleared. I have now worked out exactly what caused the problem and have explained how to deal with similar events in future.
The stalled update included a package called apt-antix, which I assume contains updated repo addresses. apt found evidence of a previous edit in the relevant file (which must have been done by some script or other) so it asked whether it was OK to proceed and overwrite the edited file with a new one. And of course nothing more happened. It just sat and waited for an answer. And my friend thought something had gone wrong, panicked and interrupted the update. That was what left the system in such a mess.
When we cleared the blockage together yesterday, the same question came up and I told her to answer 'Y'. I've also given her a general lesson on answering questions posed by synaptic. She already knows not to OK an update that contains substantial deletions because that's not normal. Now she knows that overwriting a configuration file is nearly always OK.
All the same, when the curfew is lifted, I shall create an account for myself on that machine, so that I can log in remotely when necessary.
The whole experience makes me all the more grateful for how much more simple updating is on Slackware
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