First off, since the oil pressure regulator allows flow (not just pressure) to bypass the engine galleries, doesn't that to a great extent remove the effect of the constant displacement pump, and in fact make the oil system a constant pressure system? That is how I always understood it to work!
Exactly correct. The only reason we use a positive displacement pump for the oil is because it's a very convenient type of pump to install in a crankcase for this purpose, driven by a cam lobe. The pressure relief valve essentially turns it into a constant-pressure pump, at least as long as conditions allow the pressure relief valve to exist somewhere in its operational bandwidth between full open and full closed.