After looking at the math a bit (including the one helpfully provided above), I think you're missing alot by ignoring the effect of the rotational velocity when thinking about this effect. The spinning rotation has a multiplier effect on the air movement similar to a how a much larger/fuller sail could redirect the air flow to produce more forward motion in a traditional sail.

[Edited] The effect is similar to how a rotational wing aircraft also produces lift that is not strictly proportional to the area of the wing (although for different reasons).