That’s an interesting dilemma. Obviously your engine is tuned very well, or it probably wouldn’t get that little bit of life after you release the key. Good job on that.
There are a couple of things that can cause this. Primarily though, the ignition switch is a known trouble point where if you turn it too far to the start position, it will cut off the other circuits.
Both the red with green stripe wire to the ignition coil and the green with red stripe wire to this voltage regulator are supposed to stay energized when the key is in the start position.
Right now it sounds like they are not.
You can try a couple experiments first. See if you are turning the key too far by just barely turning it and when it starts to crank keep it there don’t go any farther.
The other experiment is to see why the starter relay is not doing its job.
Is this a new wiring harness? Or original? If original, remove the brown wire from the “I” post on the starter relay/solenoid, and apply 12 V to it.
You can run a jumper wire directly from the battery positive to the brown wire. Just don’t leave it there for longer than it takes to run this test.
Once you have power to the Brown wire, immediately go in and turn the key to start and see if it works normally.
If it does, you have a defective starter relay.
And maybe a defective ignition switch too!
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