Dave,
Your logic is impeccable, if the fin were the only thing flying, but the body and rotor are there too. The total power needed in the side flight goes down by an enormous factor as the wind speed increases, usually the first 10 knots equals about 10% increase in hover gross weight. The main rotor does not care where the wind is from, it starts gaining performance at 8 knots or so, even in a tail wind. Since nobody allows increase in gross weight to account for wind, the net gain with the fin is over 7.5% (2.5ish from the 10) in hover, of course because of the wind!
Sasless, the Chinook family is a hurtin configuration in yaw, because the twin rotor provides no natural yaw stability. the tail rotor on a single rotor config is powerfully stabilizing, the rotor area is equal to a fin of 4 to 8 times that area, so little vertical fin is needed for stability. The exception is a fan-in-fin, which has little addition to the yaw stability, and so needs extra fin area 9see the end plates on most EC machines with fenestrons, and now on Comanche).