Have you inspected the wiring sockets terminals of the system relay or exchanged that relay out?
There is a ground wire coming out of the socket that has to be in great shape and I believe it fastens to the body of the car inside the cabin. So make sure the braided ground strap from the engine is fastened nicely to the firewall too!

This is the same relay the powers the fuel pumps. The relay is made with two halves inside that is also controlling the other ignition module that is located down under the glove box with said relay.

The module out by the battery, you changed, is only an electronically triggered (by the ICU) electronic relay module, to fire the ignition coil. It is only a middle man that handles a single function of switching higher current with larger semiconductors.

It is rarely is prone to failure but was packaged outside the main ICU due to its own internal heating and for ease of replacement. It may have made it more affordable, for all envolved, as we do not have to buy the far more expensive complicated ICU package.

After the ECU sees the crank signal from the CPS that the engine is turning over a susession of operations click into place. The mechanical relay and electronic programming of which the latter appears to be very reliable. It's mechanics and wiring damage most due to the a weather environments we have battles with.

If you have to you can open the system relay and tie a string around it and manually make it close to see if you get spark when cranking. It takes " I guess it working" out of the equation but you cannot leave it that way because it takes a "safety guard" out of play and you do not want guess if its working.

In fact, I think it should run if power is getting to the ECU from the ignition switch. That would be a real heads up if it did not! May go back to fuses or a bad wire for sure.

Keep plugging or unplugging you will find something or scare the bee-geez out of it! (:-)

Phil