That’s true for an ideal PD pump before the relief valve opens — it displaces about the same volume per rev regardless of grade with thin oil having a little more slip.
OK, we are on the same page there.
But in a real engine, thicker oil builds pressure faster, opens the relief sooner, and more flow gets bypassed back to the sump instead of going through the engine, and these effects are even stronger while the oil is cold, making it take even longer to reach the temperature which was the begining of this debate.
Which real engine are we talking about though? And how cold? This is all wildly application dependent. Also, many applications (like both my HEMI's) have heat exchangers that work to increase the rate at which the oil is heated, while also limiting the temperature that the oil can reach.
Thicker also moves slower through clearances (even without a bypass, there is a differential effect of viscosity on different clearance dimensions), so less oil reaches some parts per unit time, and suction losses or aeration can limit delivery when cold. So viscosity absolutely affects how much oil actually circulates in the engine, even if pump displacement is constant.
If the oil is leaving the pump, it's in circulation. Things like rod bearings will draw less heavier oil from the galleries, because they need less of it. So again, assuming the pump is not on the relief, certain clearances will just experience higher pressure for the same amount of flow, which we see in the difference in pressure observed at the gallery right after the pump. But again, this is predicated on the pump not being on the relief. If the relief opens, then we have less volume in circulation, but as I noted, this is very application dependent, and there are some engines now, like the Pentastar, that don't use a conventional PD pump, but I think for the sake of this discussion, we keep it focused on PD pumps and behaviour of the system relative to that.
As
@johnmyster noted, this is a pet subject for
@ZeeOSix, so I'm a bit surprised he's not in here already, lol.
@Shannow has written some considerable material here on bearing design and lubrication as well, though not in more recent years, as he spends less time on the forum than he used to.