Following Richard's instructions I've managed to figure out and repair the problem.
To eliminate Q1 as a suspect, I lifted the bottom (source) leg on Q1 and measured voltages with it detached from the board, I got the expected 12V or so, so I put the leg back again. I also had the expected readings of 14V and 80V on the top (g) and middle (d) pins respectively.
This meant that it had to be something further 'downstream', and that I should look at U1, U2, and U3 next. Richard suggested snipping the source pins to each of these chips in turn to see what made the short go away, once there was no longer continuity between the ground and the 12V rail, then that should be my culprit.
First I went to U1 and cut pin 8. It made no difference: I was still measuring 0Ω across the charger input ground and the 12V rail. So next I cut pin 6 on U3 (the source for U3 and U2) and this did it! I no longer showed a short across GND and the 12V rail.
By luck, I had a spare U3 chip left over from my Mouser order, as the board had already come fitted with U2 (this was part of fixing a 'bug' in this version of the board), leaving me with one left over (U2 and U3 use the same chip). So I carefully removed the one on the board and replaced it with a socket. Richard suggested using sockets for these chips to make replacement easier, as replacing the chips is quite tedious and time consuming.
U3 now sits in a socket
Obviously there was still the matter of the snipped pin on U1: For this, I just bent the snipped ends back together again and used a blob of solder to effect a repair. This done, I fired it all back up again - through the light-bulb at first - and I had my little green LED back again

It's all now up and running again, charging as it did before.
Thanks a lot to Richard and Gary for being so patient in helping me troubleshoot and fix this! It's been frustrating for everyone, but at least we have a basic troubleshooting path for figuring things like this out. I think the biggest lesson to come out of this is to use sockets for U1, U2, and U3, as this would have made figuring this out and fixing it so much easier. Perhaps three sockets should be added to the parts list, and the next version of the instructions amended to recommend that these chips be unpluggable in this way, as that could greatly simplify troubleshooting!
As for how U3 go damaged in this way - I think Richard mentioned that U3 might be prone to damage from static from the EOC jumper, and when I was trying to get the EOC cut-off to function I did a lot of fiddling with that jumper, often just shorting it with a screwdriver end as I tried to figure out what was going on, perhaps that's what killed it off...
Incidentally, I am still back at this original issue, which is that the whole thing only seems to work with the EOC jumper on. Any thoughts on what could be going on here would be appreciated, as it needs to be able to turn itself off once it's done...