A thought has occurred: is it possible for the power supply to fluctuate so as to a) cause random shutdown and B) fail to charge a battery and yet c) start the machine up most of the time?

I have another on which the casing is no more, and the plug is busted, but the innards appear to be OK. I could transplant these innards into the intact one if this was thought a possible culprit.

I went to the length of transplanting the other logic board into the machine last night. It won't even see the battery.