Fuel issue caused by electrical?

So…I'm confused. Lost really. Vehicle is a 1988 Cabriolet. I've had an issue with the fuel pump not priming. The car will start, run a short time (20 seconds or less) then sputter out and die.
I bought the vehicle running, terribly, but it was missing the front motor mount, and the passenger's side mount was trashed. I replaced those, drove across the street, put in some fuel, drove back and turned it off while I cleaned up the work space, when I went back out, it began the previously mentioned symptom. Now the only lead I have on finding the issue is at the coil. When I leave the key to the on position, and remove either wire from terminal 15 and try to put it back on the fuel pump will prime just as I begin to put the wire back on or just as I get it off. (Discovered this by accidentally leaving the key to the on position) But not when the connector is completely on.

Parts I have replaced AFTER the symptom started:
Both fuel pumps, and filter
Fusebox (had some melting and needed to be done anyway)
All relays at the fusebox
Coil ground strap
Coil
Ignition Control Module/Unit
Wire connections at the coil  for terminal 15
Cleaned connections on the coil itself


Where do I go from here?

With the fuel pump relays mine has never primed before starting and I've got a 1988 Gti Cabriolet. It seems some relay's prime and some don't, I have a No.62 relay fitted to my car so normal for not priming.
Have you replaced the fuel pump relay? You can short out the fuel pump relay to get the pump to run all the time, either remove the fuel pump relay and get a bit of wire between the 2 bigger connections or take out the indicator relay and put that where the fuel pump relay should be, this will make the pump run so you can see if that's the problem.
 

Thanks! Ok, so that would eliminate the priming concern.  I am running a No.62 relay, and I did replace it with the same from an 88 cabriolet that runs, just to see. Though I get the same symptoms. I assumed electrical since I wasn't getting a prime, and now with a second pair of hands I can hear it running during the short time the car is running. It will also constant run with a jumper in place of a relay. Pumps sound healthy, and are both brand new. When the motor dies, it gives every indication of fuel starvation.

the reason it primes when you disconnect and connect the coil is the coil is sending a single spark pulse when powered on/off, this will then trigger the fuel pump as it thinks engine is cranking. but then when no further spark pulses come in it kills the pump again

I suspect this is the same reason some kjet cars prime and some do not.

if the car will run with the relay socket bridged this suggests to me the relay is the problem, open it up and look for burnt contacts or loose/dry/cracked solder joints