Green pin of ATX MUST BE CONNECTED TOGHETER WITH GROUND! in order to the PSU turns on.
If you want to use ATX PSU like a regular Amiga brick, put a switch in the PSU and use green and a black wire to turn on/off.
If the PSU turns on only for a while, it means you have a short-circuit on one rail. Normally, Amiga's shield need +5V connected (not a permanent one, BTW). Said that, logically you short the +5V and ground via Amiga's shield wire.:crazy:
ATX permanent +5V don't be needed to plug anywhere, so don't bother to plug it (simply cut it out). The same for 3.3V (orange) wires.