This seems to have become a "Rebel" vs "Yank" issue. The fact is that the C.S. couldn't even produce metallic ammunition. They threw away captured Spencers and Henrys as soon as captured ammo was exhausted because C.S. Ordnance could not make the cartridges or make the machinery to make them.
A bigger factor is the population numbers. Someone (Catton?) said that the South could have an army or could support an army; it could not do both.
Practical auto weapons awaited the development of smokeless powder, both because black powder fouled too much and because it did not have the necessary pressure spike to properly function an auto weapon. So even if some southern Browning or Maxim had invented a machinegun he could not have gotten it to work very well without smokeless powder.