An issue with the oil pumps is their construction. The housing is cast iron while the bottom plate is a softer metal. After a while, the heat/cool cycle causes the bottom plate to warp and oil, under pressure, is forced out thru the opening, causing a drop in pressure.
The lead gasket was designed to absorb some of that distortion but who knew we'd still be driving these cars 60 years later. The solution is to ensure both surfaces are flat and that a gasket is in place. If you use paper, make sure it's of a material that can stand up to oil under pressure. The last thing you want is gasket material plugging an oil gallery.