Tenrocca,
We started the LS-1 in our Cobra for the first time last week.
We rebuilt the motor ourselves, and lubed everything thoroughly during assembly.
We didn't see much
oil pressure on the guage while cranking, but as soon as she fired - loads.
Priming the pump is a good idea if you have the time and materials, but I believe it's a luxury.
If you consider how "Primed" an LS-1 is when it's been sitting idle in your garage for 6 months - I think you can safely assume that the pump will prime itself.
From our experieince - you aren't doing enough revs while cranking to see much pressure - mind you it looks from the video as though your battery has more grunt than ours as your motor is turning over a bit faster than our was - so you might see "some" pressure.
Note to self - buy a new battery !