I honestly think a motherboard recap might help your IIcx power up correctly with its own power supply too. I've seen all kinds of weird behavior seemingly blamable on the power supply that was really caused by leaky caps on the motherboard screwing with the soft power circuit. I know on the IIci several of the electrolytics that leak are right over by the power button circuitry. I bet the IIcx is the same way. By hooking up an SE power supply you're probably bypassing that part of the motherboard's circuit...

Of course, on the other hand, maybe the power supply really is just bad. Who knows...