This is a problem located on the logic board. It's more than likely caused by a bad capacitor or two.
Notice that if you remove the logic board from a 5260, 5500, etc, the shell auto-magically turns on and fires up the display. The logic board actively pulls some signal low (presumably) to keep the Mac turned off if it's working properly. They do it this way so that if the soft power circuit fails, it will just always turn on by default. Many Macs with soft power exhibit this behavior with age as capacitors begin to fail.