The OP does bring up a valid point- when we rock the bike left to right, we are transfering ARM work to the pedals. Someday somebody will create a handlebar powermeter to measure work done by the arms. When that happens we will see that indeed some portion of the arm power is wasted on overcoming the gyroscopic forces of the wheels.
However for all intents and purposes everyone is limited by their legs tiring out. Nobody is particularly limited by our arms giving out. So how much power is wasted by your arms is mostly academic (to be fair, arm power also has to be replenished with food, so it is not completely academic)