The fact that the car runs on aerostart means that a lot of things are working correctly. The distributor is correctly wired, the timing is near enough, the ECU knows what is going on and is attempting to run the engine. The injectors clicking mean that it's at least getting enough information to run in a limp mode so the problem is not your ecu.
Have you tried bridging those terminals while cranking the car?
But there are two different ways of triggering the fuel pump, depending on whether your car came with an airflow meter or a MAP sensor.
If it came with an airflow meter (like the early ae92 GT-Z with the supercharged motor) the fuel pump won't operate without the airflow meter giving a signal (it's actually controlled by an output on the airflow meter). If you move the flap by hand it might trigger the fuel pump.
Map sensor equipped cars run off the ecu. Upon turning the key to ON will prime for a second or three then stop until the ecu receives a signal from the distributor (when the engine is rotating), then start again.
Then there is the possibility of having made a dumb mistake with the fuel lines. The correct way is tank-pump-fuel rail-regulator-return. Disconnect your fuel return line and see if there is fuel being returned to the tank.
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