I may have said it already but I'd guess that the biggest reason there were no simple automatic weapons during the Civil War was no one had designed one. No complicated ones, either. But none of them are really simple.
Technology was not all that primative in 1861. Firearms were made on machinery that came from corporations still doing business today. Interchangeable parts (for firearms) had been around for fifteen or twenty years. That was done for ease of production, not so your local hardware store could carry a supply of parts for something. But as has already been pointed out, there was still a lot of ground to go over before workable automatic and even repeating firearms would be available. Probably the ammunition but the early automatic weapons were quite complicated and some models adopted by some armies even after 1900 were not particularly successful, which could even be said of some weapons designed an adopted 30 years later. By then, of course, it was usually a matter of design (poor design, that is) that accounted for poor performance on the battlefield.
As it was, the (probably) most successful automatic weapon, the Maxim, appeared less than 20 years after the end of the Civil War and in fact used black powder cartridges at first. But I'd also have to say that "simple" automatic weapons would not appear for another 60 years after that. Even a muzzle-loading artillery tube is not as simple as it looks. True, the design is simple but the metallurgy is critical or the gun will fail. Oddly enough, it tended to be the more advanced artillery weapons of the Civil War period that tended to fail more often and by fail, we mean blowing up. But that was probably often due to the gunners attempting to get extreme performance from their guns. Remember than when you are handloading your own ammunition.
Shoot low, sheriff. They're riding Shetlands!
Underneath the starry flag, civilize 'em with a Krag,
and return us to our own beloved homes!
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