Your ECU is definitely faulty. (And, yes the turbo ECU (ECCS) does "prime" the fuel line by running the fuel pump for approximately 5 seconds at ignition ON, and then resumes full fuel pump operation as long as it's getting a CAS signal that the engine is turning.)
The ECCS has complete control of the fuel pump relay. IF the fuel pump is operational, AND it's wiring is operational, AND the fuel pump relay is operational, THEN the fault is in the ECU. At ignition ON, there is no requirement other than power for the ECU to "prime" the fuel system.
There are other "markers" that your ECU is faulty - Injectors firing with no CAS signal is the biggest, and the LED is another.
1974 "early" 260Z - L28ET - MS3X full sequential fuel and spark, COP=Ford 4.6 coilpacks w/ Nissan Ignitor, diyautotune chopper wheel - FMIC - T04E .63 A/R - 400cc Injectors (too big!), 14point7 Spartan2 Wideband - C/R 5-speed - Coilovers - Big Brake Kit up front, Porterfield friction all around. Mesa, AZ