Then there's the old, and exceptionally hard to kill, theory that every American man is, by birthright, God, country, Mom, and apple pie, a sniper of incredible skill who can place repeated shots on a man-sized target at 1,000 meters all day long.
Mike, I would agree with this entirely, except, no REAL AMERICAN measures the distance to his target in meters!!!

The answers to the OP are COST, complexity, fragility, and general bigotry.
Remember to look at things in the mindset of the era, and NOT with 20/21st century "common knowledge".
Along with the already mentioned cost of the rifles, ammo "wastage" etc, there is the fact that the military arm was still considered to be an IMPACT weapon. Bayonet or buttstroke, the rifle had to be able to dish out, and take these. This is something NO lever gun design was as good at as "regular" rifles and muskets.
Lever gun cartridges didn't have the power (=range) you could get in a single shot until the later 1880s. And by then, technology was feeding itself, various improvements upon improvements, some dead ends, and some resulting in state of the art guns that haven't been significantly improved on in 100+ years. Basically, by the time there were lever gun designs that might have been suitable for military service, there were other designs, (notably bolt actions) that were even better suited.
the early lever guns had tremendous firepower compared to muzzle loaders and early breechloading single shots. But they were also complex, expensive and not nearly as rugged compared to those same single shots.
and the Army of the day was not yet wedded to the idea of the individual soldier have the greatest firepower (or even the best weapon) possible.
In the immediate post civil war period (and for some time after) a large portion of the Army's troopers were illiterate, or only semi literate. They were not the educated highly trained professionals we associate with our armed services today.
Also remember we are talking about a rough 30 year period between the 1860s and 1890s where there was no pressure to fight a foreign army driving improvements in arms. We were still in the process of switching over to the new, high tech "smallbore" (.30cal) Krag and smokeless powder when we ran into the Mauser in the Spanish American war. Where we learned that while the Krag was a good rifle, its wasn't the best combat weapon possible, and we adopted what was essentially a Mauser system rifle a few years later (1903).
By this point, the lever gun, even one shooting the same rounds as the bolt gun, simply didn't stand a chance of being considered.