I would quibble with your characterization of Civil War-era firearms. While muzzle-loading rifles could only fire a few rounds a minute (not, importantly, a few minutes per round) a rifled musket like the 1861 Springfield or 1853 Enfield was accurate to about a quarter-mile and dangerous far beyond that - the Enfield's maximum range is given as 2,000 yards, which is well over a mile. A Minie ball is also terribly more destructive than an arrow, which means greater stopping power per shot fired. Then there's the issue of training. Both the Union and Confederate armies needed a lot of soldiers, and fast. Learning how to aim and fire an arrow is much more time-consuming than learning to load, aim and fire a rifle. Simply firing a bow is also more physically taxing than using a rifle, an issue of no small concern to an army that walks everywhere it goes. Finally, I would argue that there was something of a cultural bias towards firearms. They had been standard issue in European armies for over a century by this point, and the infrastructure to continue producing them, along with gunpowder and bullets, was well established. Arming troops with bows and arrows would have been taking a step backwards, technology-wise, and required a major shift in resource allocation. Speaking of technology, while most troops carried a rifled musket, the Union Army also issued tens of thousands of brand new Spencer repeating rifles, which had a 7-round magazine, a rate of fire of 15-20rds/min, and an effective range of 500 yards. Some soldiers also purchased Henry repeaters, with a 16-shot capacity, on their own.

Indeed, as /u/ hatari_bwana has already so eloquently and correctly pointed out, civil war guns were anything but underpowered or inaccurate. If the movie presented them as such, they are incorrect. Even obsolete smoothbores could deliver lethal, aimed shots out to about 100 yards.

Despite the time it takes to reload a civil war era rifle, they are still superior to a bow on the basis of portability (rifle, maintenance supplies [smallish], pellets [small], black powder [small]) and durability*. I say durability with an asterisk mainly because, truly, it was not necessarily their immediate durability that made them so efficient, but their ability to be recycled or easily fixed. Furthermore, a rifle does not require the strenuous work that a bow does, and could therefore be slightly more efficient over time. Oh and I suppose ridding itself of the various human variables (strength, placement ect.) weighing down a bows accuracy rate probably helped too.