You're only partially correct. The issue is that while a lesser weight arrow takes less energy to propel forward at a faster speed, because of the reduced mass things like wind friction, shelf friction etc will have a greater effect on it because less mass=less inertia; Also, the arrow can only go as fast the limbs are traveling forwards, and the fastest possible limb travel would occur if you dry-fired your bow, which of course you do not want to do.
A similar, albeit not totally the same, example to think of would be that lets say a given arrow drops 1 inch at 10 yards, 2.5 inches at 20 yards, 6 inches at 30 yards, and 20 inches at 40 yards. If you were to plot these data points on a graph you would find the data is not linear but rather exponential, as the arrow is continuously slowing down due to air resistance at the same time it is being pulled towards to Earth due to gravity.