Yeah, I'm thinking that because it boots OK with the keyboard power button, but not with the rear power switch, that's a strong clue it's a capacitor problem. I see the keyboard power button directly grounds the junction left of R43, but rear switch grounds it through C6. And then it'll only stay grounded for a brief time, whose duration depends on the values of C6 and maybe C12 and C13. Keyboard power button also doesn't touch the power-off circuit (that whole section at lower center), but the rear switch does, so I suppose anything in that section could be bad too, including the caps or any of the ICs.
Once booted, the Finder's shutdown command works OK, so I'm guessing that power-off circuit is fine. So my bet would be on one of the three caps in the power-on circuit, most likely C6. If its capacitance has fallen to significantly less than its rated value, I think that would produce exactly the behavior I'm seeing. It would charge to 5V too quickly, bringing the R43 junction high again before the PSU bootstrap through D6 could be established. So it would just click on and immediately off again. Although now that I think about it, that wouldn't explain why pressing *and holding* the rear power switch works. Oh well.
Anyway, recapping, here we go! Fun and excitement will now commence.