The square 4-way plug was introduced for the current-hungry P4 and is nowadays also used on Athlon64/Opteron mobos (previous Athlon generations often suffered from 3V/5V rail PSU overloads) - you don't need it at all.
I'm pretty sure the pinout of your adapter is wrong: what you are saying (short flash of the LEDs, fans making some turns) sounds extremely like a short. The PSU detects overcurrent on powerup and shuts down again immediately. Does the setup work with the drives attached, the ATX->AT adapter attached, but no mobo? Then again with the AT->Amiga adapter, but no mobo?
