In reply to elsewhere:
Thanks, that makes sense.
I just tried it again this morning. The battery was in the bike and hooked up to the tender by the fixed cable. The tender was saying that he battery was fully charged. I disconnected the tender and the voltmeter read 12.4V across the battery terminals. I turned on the ignition and left it for no more than about 5 minutes, after which it read 10.4V! I tried turning the headlights on and it seemed to be struggling to trigger the relay on the lighting circuit. I reconnected the tender and it was straight back in to charging mode, and has been for over an hour now, which seems excessive for the load I asked it to support for just a minute or two.
That does seem to tally with the process you describe, where a bad cell appears to charge to the full voltage but actually has only a tiny amount of charge capacity, which gets fully depleted very quickly.
There's no question that the battery is no longer serviceable, and it does look as if the tender is doing its best in the circumstances so can still be trusted. Which is nice.
I did have another battery tender before the current one which I'm pretty sure did kill a battery. That brand seems to do well in group reviews online but I'm afraid I couldn't bring myself to buy another one from them, or recommend them to anyone else.