Yes, the Confederate Army did have a few Whitworths and they did very well with them but they were far from common and did not meet the perceived needs for the sniper role in the Confederate Army, especially in the West where they were practically unheard of. A fair number of standard rifle muskets were more than capable of the accuracy needed for snipers in the days before optical sights were available and a man with good natural shooting ability could more than satisfactorily use one in work of sniper. As you are well aware, scope type sights were rare and inordinately expensive at the time of the American Civil War and so delicate that they couldn't stand up to any kind of serious civilian use, much less that required by the military. Open sites were all that were successfully used by Snipers at the time and even exceptionally sharp sighted people would not be able to make accurate use of them beyond, say, 500 yards - few could even do that. Remember that the longer range rear sight of the Enfield was made for volley firing at long range, the targets being of at least company, even regiment sized targets.
The Sharps Rifle was a wonderful contraption and it did reasonably well in military service. And Berdan's men were almost all capable marksmen with nerve and they could serve as snipers but they were far better and more heavily used in the role of skirmishing and as flanking troops. As such, most of the accurate fire most were capable of was done at only slightly further range than a line infantryman was capable of with training and natural ability, both of which Berdan's men had in spades. For skirmishing, the Sharps, as long as they continued to use them, was a great rifle is relatively expensive to maintain. It's reasonable accuracy, especially at the 200 yards and less range, gave Berdan's men the ability to fire relatively fast and made the most of the men's ability to stand and fight as well as to shift their positions rapidly that their training for these duties taught them.