I’m just thinking out loud here, but maybe there are some mix-matched components. It’s only a guess.
From your description, I think that you have
oil pressure at the pump, it’s just not showing at the rocker arms. It can take a while to work it up there. Perhaps a remote
oil cooler adapter or
oil thermostat could be installed backwards? There are flow check valves in those. If you can prove the pressure at the filter (just remove it and spin the pump by hand, it should flow) you will be ok with the pump. You can delete the oil cooler for trials. I have primed reluctant pumps by filling the crankcase above the pump to completely submerge it, turning the pump by hand, then draining oil back out to the correct level. Again, it sounds like you have pressure, it’s just not showing oil at the top end. Hope this helps.
Jake