I thought it could be capacitor-related, especially since sound is also not working (the soft power off issue is more annoying, however). I have now disassembled the IIci so I can take a good look at the logicboard. On further inspection, I noticed four problem areas:

PRAM battery looks a mess, obviously needs replacing: (I already have the replacement battery)



This doesn't look too great:



Some minor mess here:



And here:



In terms of capacitors, they physically look good, no goo or bulging in sight.

Seems like the best thing to do is wash the logicboard, put in the new battery, and see what happens.