The missing wire was formerly used to provide -5 V power to the motherboard, that was only used by old boards for ISA slots, any board with no ISA slots has no need for -5 V power, and any PSU designed since about 2005 won't even be producing -5 V power so it has nothing to connect to that slot.
How old is your old computer? It would have to be really really old for this to be an issue for it. Do any fans spin at all? You may have a DOA PSU, but wait for the rest of the parts to get their so you can check it in the system.