I was under the impression that hyperventilating is more about the reduction of CO2 in the lungs, rather than saturating the blood with oxygen.

It's basically to fool your body into thinking you've got enough oxygen (since it's C02 buildup in the lungs that triggers the out of air signal to the brain, not the lack of oxygen).

Hence why hyperventilating for, say, free diving can cause one to black out without triggering the suffocation reflex as one returns to the surface of the water.