There is a redundant oil pressure circuit to the fuel pump relay. If the ECM is receiving a pulse from the distributor it turns on the fuel pump by activating the relay. Or if you have oil pressure, that will activate the pump directly. Not all swappers use this type oil pressure switch. Later year 4.3 Vortec's didn't have that circuit either, but a 96-97 would have had it. It's not necessarily needed. If you aren't getting an ignition pulse no need for fuel. Prior to computers GM used it as a safety feature. If you didn't have oil pressure, you didn't need fuel. Like if the truck is upside down.
It sounds to me like something from the ignition switch to the ECM is wired incorrectly. Sounds like you have 12V in run, but not in start. Check fuse #10 ECM/IGN to be sure it has power in both run and start. I've run across custom harnesses that failed to provide power in both positions. Usually when someone is building a harness they check for continuity in run, but don't always check it in start. Basically all pink wires are hot in run, but not all are hot in start. This is done to prevent surges from damaging sensitive electrical components.
Or it could be a problem with the ignition switch.