I think it's cured, so I'll close this topic.
Many thanks for the advice.
I opened it up, and although nothing seemed to be loose I made sure things were pushed fully home. What it may have been, though, is that there's a sheet with a coppery coating that goes over the circuit board. It's anchored at each corner, but it had come loose - presumably during the hammering the synth had before I got it - and I wondered if that might have been shorting something. As an experiment I removed it completely and put the synth back together.
It now seems to work as normal. All the patches have gone, but it powers on correctly every time, and all the select and edit options work. I have one of my patches written down by hand, so I'll put that back in and do some experimenting like I did before to get more useful sounds - and also sample it this time. I probably won't bother looking to download the original patches.
I'm assuming the sheet was just protective - but please tell me if it's something vital.
It doesn't seem to be.