@siriushardware: The plug was the first thing I checked, and I've used two different cables (composite and scart) so I've plugged it in and out quite a few times. The new shifter chip made the machine run again. I don't think that you can fry a shifter by plugging the cable upside down.
Sorry, I should have clarified: Plugging the 13-pin monitor plug into the ST upside down will TEMPORARILY produce those symptoms you described. But actually, you can cause damage by either plugging the plug into the monitor socket upside down or unplugging it upside down while the monitor and the ST are both on. Guess how I found that out? It didn't blow the shifter but it did blow one of the 74xx gates attached to vsync or hsync, I forget which.
My advice is never to connect or disconnect things while they are powered.
...It's a pity I don't always follow my own advice.
Good luck with finding your problem. I discovered recently that Freezer spray is not very cheap, so if you don't have any already, don't buy an enormous can of it - just the smallest can you can buy will be enough to hunt for this particular problem.