I may have to try this in an attempt to resurrect the dead board in my TerraMiner as well. As you recall, I have one machine that is only running on one board as well. It kicks on for like half a second and then immediately turns off. So, why do you need to run it on a car battery to "fix it" (or, rather, from the sounds of it, to blow up the one bad mosfet or chip that it preventing the board from working and then the rest of the board will work) like you explain? If you try running it off of the power supply, is the board just telling the power supply to stop supplying power because of a short, but the car battery doesn't care and will keep supplying power until it blows up? That doesn't seem to be my experience though as when my machine is running, I checked the voltage at the big power connectors to the board and I was getting 12 VDC to the dead board, it just was still not turning on and working. I would think then that this would be the same result if I apply power to the dead board from a car battery - it will try to turn on for half a second and then turn right off to try to "save" the board. (Which, in actuality, is still ruining the board since I can't use it either way! lol...)
It sounds like your problem may be different if you are reading 12V at the big connectors. Do you have any orange lights on your power supply?
To answer your question though the reason you need to use a car battery (or a monster bench power supply) is because you have a short in the mosfet (source to drain is most common). This short causes the board to draw too many amps from the power supply and the overload protection kicks in. I believe the terraminer has a 1100 watt power supply so if we assume that one channel is drawing all 1100 watts then that's roughly 91 amps. If you are going to blow out the short you need to supply more than that. It's possible a supernova 1300 could supply enough but the easiest and cheapest thing to do is just get some #12 primary wire and hook it to a car battery. Most car batteries are capable of supplying in excess of 600 amps at 12V plus there's no overload protection so even if you draw all 600 for a few seconds it's not going to kick off. That will be more than enough time to overheat the shorted mosfet.
Now on to what I think your problem is. I suggest swapping power supplies to rule that out as a possibility. If you take the retaining screws out from the back of the terraminer and depress the release clip they slide right out. Just switch the 2 you have and see what happens. If it follows the power supply then that's bad. If not then you have a board problem. I still suggest trying the car battery approach to rule that out as well but it's likely something else failing on the board. Usually catastrophic failures can be spotted visually. Remove the board and carefully inspect all hardware. It's also possible that the board is not detecting the water pumps. You can try swapping the pump cables from the board that works just to see if it will run. DO NOT start mining like this because you will overheat it almost instantly but it will at least help you narrow down the issue.
Let me know how all that works and we'll move on from there.