Good news! It seems my Raspberry Pi isn’t dead after all, though the problem is a little odd.
I ordered some new SD cards so I had some spares, and out of interest put a new copy of Raspbian on one of them, put it in the Pi and it booted happily first time. Yet it wouldn’t boot from the old SD card that was fine in another Pi! I’m not going to complain, but I can’t claim to understand why…