That could be a check valve. The pump is smooth and quiet. It's not cavitating and it loses pressure immediately with the trigger pulled, so it's not a restriction of supply or a leak allowing air. There's a check valve right at the pump in your video. Disconnect that line to it and see if water is coming out from it. That one is probably your weep, and there will be at least one more at the boom you need to check. Once you either find or eliminate those as the problem, it's time to open up the pump. Check all the valves first, make sure there is no debris under the poppet and the o-rings and backup rings are good.
The pump is able to produce 1,100 psi when it's in unload mode (closed cycle) leading one to believe the seals are fine. That would just leave a restriction downstream from the pump to the trigger because of a pressure drop when the trigger is pulled. There's enough flow to activate the unloader, but not enough to operate at the desired pressure.
A simple test would be to install a new hose and gun at the unloader and test it in the equipment room by squirting it in the drain.
Doing that takes everything out of the loop right up to the pump, minus the unloader.
It would be interesting to see how the unloader has been connected as well because it seems to be producing desired pressure when the trigger isn't pulled, and regulated pressure when it is pulled.... weird.
If you look at how clean his pumps are, and then the fresh belt shavings under this pulley, and his revelation of the problem with a belt at another pulley, it seems like a new problem has recently developed that's making the pumps work harder than normal.
Possibly even a blockage at the inlets?