If you feel there is misinformation being presented, please take the time to address it.
Most oil pumps are positive displacement, which means that, as long as the relief is not engaged, the same volume of oil is being displaced per revolution regardless of the viscosity. There are some minor caveats, like pump slip with thinner oils, which results in a slight reduction in volume moved, but assuming a perfect pump, a PV pump doesn't care what the viscosity is. Yes, as you touched-on, this means that there's an increase in pressure in order for that volume to remain the same with the heavier viscosity fluid.