Since everyone else wants to kiss the trapdoor Springfield ring, I'll be the one who doesn't; these were terrible rifles. Sure, they look nice, they have that cool breechloading action, they don't recoil much with certain black powder loads, but there were much better rifles available at the time, specifically the Remington Rolling Block, that could have been chosen as an infantry weapon.
I mean, let's look at it this way: in the years after the Civil War, who was the US most likely going to have a conflict with next? It wasn't going to be a European country, they all figured it out by then that they could not fight a war in the Americas against forced as well armed as they were. Mexico? Mexico was more interested in building railroads and telegraph lines than fight a war with another country, especially after they had just dealt with the French. Thus, there was no one other than Indian tribes that the US was preparing to fight.
Yeah, Indians, who tend to prefer close range combat with melee weapons and maybe some gun fights for those who had the arms and ammo. A lever action would have been a much better rifle at the time for most troops, maybe you throw in the Springfield for some designated marksmen use for longer range shots. The biggest benefit of the lever action rifles tho was the Army could have deployed troops with rifles and revolvers chambered for the same round in .44-40 or maybe .45 S&W. You can't tell me that Winchester and Smith & Wesson couldn't have come up with designs in the early 70s to create a rifle and revolver combo that would have beat the snot out any competition.
But the goal at the time was to not spend much money and the result was ill equipped troops for 40 years.