I don't know of an oil that is 10/40 that has zinc in it. You would have to use an additive, or at least I would.
The only issue here again is with the 0 blurp in oil pressure. Everything else is fine. IMO
I would Run it the way that it is. If you KNOW that the oil pressure blurps to 0 than I would try adding more oil and see what happens. I would still think it would be the pickup causing the problem. Unless you blew up your other engine and BIG CHUNKS of metal got in the pump or oil passages..........I always use a NEW pump no matter how many miles were on the engine.. Cheap 50-$80.00 insurance. JMO