Shawn, building on your well laid foundation:
Since the lift is the same, at 1' and 100' , the Relative Angle of Attack that each blade section requires is the same. Since the induced velocity to produce this lift is lower near the ground ("mirror image" helicopter is generating a gentle descent flowfield) this means that the pitch goes down to maintain the same Rel AOA. So the same lift force (and hence total pressure) is generated from a lower collective position. Thus a lower induced torque, hence power, is required.
