The "Henry" of 1860 was used a little during the Civil War, but was a .44 Rimfire.
They were also used on the plains, along with the early 1866 Winchesters.
Even the model 1876 Winchester was sneered at, by the buffalo hunters who were mostly using single shot rifles by Sharps, Remington and Ballard. The Winchesters all had very light bullets(for calibre) shot from guns with slow 'round ball' twist barrels & were said to not be powerful enough to kill a buffalo. During the same time, mounted "hunters" military and otherwise, were shooting buffalo ("running" buffalo in was called) from horse back with martial pistols & revolvers. I suspect one of the .44/40 rifles, like later 1866's and the .73's would also have worked well from horseback.
The Military perspective was that if soldiers were armed with multishot weapons, they would "waste" ammo, thus cost too much.
The cost factor was also the reason that the military used copper cased ctgs. in the .45/70 long after brass cases were on the market & being used almost exclusively by the civilian hunters.