You can use another IIcx power supply to test or one from a IIci, IIvx, Q700, C650, Q650, PM7100, IIsi (but out of the case). They all use the same connector, so if you have access to a good power supply of any of those types that might make your diagnosis easier.

The behavior described would make me suspect the four caps in the power supply start-up circuitry on the logic board. But at this age, they all need replacement.