As to why the US didn't ever adopt lever action rifles: at first, money. With the end of the Civil War, the US Army realized it needed to upgrade to breechloaders. Some ( including lever action Henry rifles) had been used by some units, mostly cavalry, but the standard arm from 1860-65 was still a muzzleloading rifle. The War Dept did trials in 1866 of many different kinds of breechloading designs, which ran the whole gamut from single-shot guns like the Sharps and Snider, to magazine weapons like the Spencer and Henry. Unfortunately, the Henry ( later Winchester) had already had critical comments made on it during the War. On Dec. 9, 1861, Brigadier Jas. Ripley of the Ordnance Dept. wrote to the Secretary of War, (in part):

....I do not discover any important advantage of these arms over several other breechloaders, as the rapidity of fire with these latter is sufficiently great for useful purposes without the objection of increased weight from the charges in the arm itself, while the multiplication of arms and ammunitions of different kinds and patterns, working on different principles, is decidedly objectionable and should, in my opinion, be stopped by the refusal to introduce any more unless upon the most full and complete evidence of their superiority. In view of the foregoing, of the very high prices asked for these arms, and of the fact that the government is already pledged on orders and contracts for nearly 73,000 breechloading rifles and carbines, to the amount of $2,250,000, I do not consider it advisable to entertain either of the propositions for purchasing these arms.

from Claude Fuller: The Breech-loader in the Service

You can see a sort of general complaint; all these new-fangled guns are costing us a lot, and they're all different and complicated: and Henry rifles are even more different, complicated and expensive. Given that attitude, it should not then be surprising that the winner of the 1866 trials was a relatively cheap and simple conversion of the Model 1863 Springfield muzzleloading rifle, of which the Army had thousands and thousands. Even better, the conversion was thought up by an armorer at the Springfield Armory, so it was "in-house", and no royalties would be due to an inventor of one of the competing designs ( though, it turned out that someone HAD submitted a design very much like the armory conversion, and would eventually have to be paid) .This design, which eventually became the Trapdoor Springfield, stayed in use into the 1890's, long after every other army in the world had gone to repeating arms. Compared to most other designs, it was not very good. But the US had few occasions to need anything good, 1866-1896.

Though the US did not adopt Winchesters/Henrys, other countries did. The Turks actually bought a lot of Model 1866 Winchesters and used them very effectively against the Russians in 1878. And some parts of the design were noticed: the tubular magazine was quickly incorporated by the Swiss into their new Vetterli bolt-action, and also by others in the 1870's. But by the time the US got around to discarding, finally, the Trapdoor Springfield in 1892, it chose a bolt-action. Because by then bolt-action rifles had proved themselves to be better in service than other designs (including lever action repeaters) being more rugged in use, stronger, simpler to take apart and clean, and easy to fire standing, sitting or prone.

There was a good more detailed discussion here as to the merits of bolt action vs lever action.