Bow to the bow
Initially, I wonder how many bow hunters were around in those days, especially in the south. It was obviously cheaper than a fire arm for poor folk, and had the salutary effect of not scaring off all the game with the first shot as guns did.
A second question is whether most riflemen were also armed with a revolver, allowing an increased rate of fire against an advancing line as they neared the defensive positions. This would cut into the rate of fire argument in favor of bows. If not, then their use at short range might have been of great advantage. Barring the existence of a considerable number of personally owned bows, the most salient factor against their use was probably one of logistics, i.e., their manufacture, transport, the weight of an arrow vs. a mine ball, etc. Conversely, unlike lead ball ammo, they could, under some circumstances, be recovered for re-use.
Remember, they passed out of use because of defensive advances, not just the advent of firearms; but the Civil War soldier had no such defense against their use. I shouldn't think that the bows to be used would have been prohibitively powerful like the long bow, nor would they be used as extreme range but only within a hundred yards or so, so the specialized training would not have been such a negative factor. Simply aiming generally at the advancing line would have been sufficient. Also, they had the added benefit of not being line of sight weapons, allowing archers to be almost totally hidden from fire behind defensive structures or terrain (tho' it's my impression that advancing troops themselves did not fire as they advanced, just artillery). If used against the flanks of the line, the commanding officer might not realize that they were being attacked and weakened due to the lack of sound and smoke, and might forego reinforcement and allow him to mistakenly rely on the decimated flanks for maneuvering they were no long capable of.
It is hard to discount the rate of fire almost 7 or 8 time greater than the rifled musket, which was also used in general fire, not necessarily against picked targets obscured by their own smoke. Let's not forget, the bowmen would be riflemen before and after their use, so they are not "dedicated" bowmen. Finally, they would be invaluable adjuncts to the rifles in night encounters, preserving the element of surprise as long as possible and sowing confusion in the enemy's lines.
