"What! what! men, dodging this way for single bullets! What will you do when they open fire along the whole line? I am ashamed of you. They couldn't hit an elephant at this distance"

-- Gen. John Sedgwick's last words.

Whether he actually said these words or not, John Sedgwick was fatally wounded at Spotsylvania Courthouse in 1864 after he disregarded or forgot warnings and walked up to a stretch of line that was known to be under fire from Confederate sharpshooters. At the time he was the third ranking officer in the Army of the Potomac after Grant and Meade, and considered widely influential.

This anecdote illustrates the evolution of thought in military circles during the war. Many officers believed that only close range volley fire was really effective, yet many also believed that long range shooting could be tactically useful. How to organize and employ expert marksmen was the subject of widespread experimentation on both sides during the war. Special Units like Berdan' Sharpshooters (1st United States Sharpshooters) and Confederate Sharpshooter Battalions were created, and many regular Civil War regiments formed smaller detachments of sharpshooters from the expert marksmen within their regiments. Some of these troops carried specialized rifles like the Sharps or Whitworth, but other sharpshooters made due with standard Rifle Muskets. Rifle Muskets could be accurate at long ranges, but were much more difficult to master due to the lower muzzle velocity and parabolic trajectory compared to the breach loading rifles which would become more common immediately after the civil war.

None of this should contradict the claim that average accuracy in the civil war probably didn't exceed that of Napoleonic (1803-1815) infantry, nor did ranges of Civil War infantry firefights greatly exceed that of the early 1800s. For the majority of civil war soldiers, discipline under fire and reloading quickly were the key military skills.

Sharpshooters and individual marksmen were a significant military resource to armies in the ACW, but this skill was not as important for the average infantryman as it would become later.

However the link you provided traces this history of the NRA to 1871. By 1871, six years later, a revolution in warfare in Europe had changed the state of the art in military small arms. The 1866 Austro-Prussian war had convinced European militaries that the breach loading flat trajectory rifle would replace the rifle musket. In 1870 the Franco-Prussian war two armies with breach loaders had faced each other and it was obvious that soldiers in future wars would be required to aim and fire their rifles individually. Infantry firepower would now depend on marksmanship as well as discipline and speed of reloading. This war was widely reported in both Civilian and military circles in 1871, with Gen. Sheridan even joining the Prussian HQ in 1870. There is nothing absurd or cultish about a U.S. military officer advocating for marksmanship training in 1871, that would have been compatible with their American Civil War combat experience and recent developments in military science worldwide.