Alright, so I think this will be my last reply to this thread. Hopefully it helps someone else in the future. I talked to a very helpful tech at the Toyota dealer. He agreed with my assessment. The fuel rail does need to be pressurized with the key in the On position (pump runs for a couple seconds then shuts off). He also agreed that the culprit could be the relays or ignition switch.

So I did some testing of the Circuit Opening Relay. A visual inspection showed no obvious damage. It's just a simple open/close relay. The voltage at the relay socket is normal. The coil gets 12V and the switch gets 12V (key in On position). So the ignition switch is fine.

Since the relay coil receives its ground from the ECU and the ECU isn't grounding the coil unless the car is cranking or running the problem lies in the ECU priming circuit.

So until I find a good used ECU for cheap, I have soldered a jumper wire to the ground side of the coil. Before I crank the engine I touch the end of the jumper to the cigarette lighter ground. Voila, fuel pressure.

It still takes a couple extra cranks to start but that's probably because the injectors aren't receiving a prime pulse to clear them of air. This is just a guess though.