Sounds like your power supply is on its last legs. Astec was notorious for producing poor power supplies for Macs, as I am sure you are well aware of (I am surprised that an Astec supply would be in a IIfx; you would figure a Sony PS would have been used, considering the IIfx was the "flagship" Mac model for a short period of time).
There could also be potential problems with the motherboard. Look for bulging caps, and leakage.

100th post, btw.

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It is unlikely to be IIfx logic board caps as the IIfx seems to have all tantalum caps and no electrolytic. Still, it seems that something is preventing the power-on circuitry from "latching". If you have the skill, I'd look at the power-on circuitry schematic (I think it's available at Gamba's site?) and try to determine what is essential to keeping the machine on, after the button has been pressed and released.

My off the cuff guess is something in the power supply unit, as there's very little to go wrong on the logic board in the IIfx.