Memory module and associated socket both with slight signs of liquid damage - corrosion, and probably short-circuiting to some of those blackened connections. Later I cleaned up both, but the same memory fault persisted. Wasn't prepared to risk putting a good memory module from another machine into the second socket, which was spotlessly clean, lest it be damaged by any other faults in the board.
Only success with this laptop was getting the power on/off switch to work - the initial problem. Saw no obvious signs of liquid. But once the machine tried to start, there were the tell-tale series of beeps that identified a memory fault. Removing the module showed some small amount of fluid had reached these parts. Fortunately it didn't reach the nearby battery connector, that's partially visible at the bottom.
Fixed the other laptop (see yesterday's Blip) with a USB connector problem, but have put off the Apple machine for another day or night.
Lovely sunny and warm day... outside.