I truly think that the bolt action would have come out as the choice as the primary action for the US military even if Europe had been completely pacifist and unarmed.
History is not always inevitable, but I cannot see any other outcome than this. But, the US Army Ordnance Bureau tried, and tried their best, to stay rooted in the last war. After the Civil War, what the US Army wanted, was something as close to the 1861 Musket as possible. The Ordnance Bureau passed up several, better, single shot designs, the Rolling Block and the Martini for two, to adopt a rifle that was as close to a muzzle loading 58 Caliber musket as possible, and still use a cartridge! I am certain the requirements group knew of lever actions, and of course, many would have been familiar with the Spencer, which was an outstanding weapon in the hands of US Cavalry, and decisive, against musket armed Confederates. But what they wanted, was something so similar to the 1863 musket, that the Union troops in Buster Keton's movie,
The General were using Trapdoors! Even in 1926, on a movie set, you could issue actors with the Trapdoor and the general public would not know the difference between the cartridge rifle and the musket!. The Ordnance Department had reached Nirvana!
By the 1890's is was getting to be obvious that the Springfield Trapdoor was outdated. By then the Russians, the Germans, the English, the French, the Swiss, had adopted bolt action rifles, the tube fed ones were in the reserves, the primary front line weapons were magazine fed bolt actions. And the US primary service arm was still the Trapdoor Springfield. When the US invaded Cuba, and 760 Spanish troops faced 15,000 Americans, the Spanish armed with Mausers, and the vast majority of Americans with Trapdoors. Even with these odds, the Americans suffered five times as many causalities as the Spanish! The most successful American action was with Gatling guns, and that lesson was ignored. Gatling guns and machine guns "wasted ammunition".
There were still lots of military, and military advisors, who were claiming single shots were still the best military service rifle, and a primary reason touted at the time, was to prevent "ammunition wastage". Logicians thought it much better for the troops not to shoot too much ammunition at the enemy, thus avoiding the vicious cycle of restocking ammunition. Obviously, a lever action would shoot too much ammunition at the enemy. It is a humorous fact that logistical laziness hampered combat weapon effectiveness. It was difficult to supply an Army in the field, and yes, it was costly, yes, but guess what, if you can't keep the food, clothing, shelters, ammunition, weapons, flowing, you will lose a protracted war against a foe who can.
One point not mentioned is how difficult lever actions are to dismount and clean. They simply have too many parts and are too complicated for not merely the average recruit, but the 95% recruit. You can verify this for yourself by going to the local gunstore and examining 100 year lever actions that were never cleaned by their owners! Too many screws and springs. I have examined a number of early military rifles, some are complicated and have too many parts, but, by the time you get to the 1880's, rifles are getting simplified. Maintenance actions are simplified. On the Martini Henry, you push out a split pin, lower the lever, and the breech block pops out. From there you can wipe out the mechanism. It does require a screwdriver to remove the trigger mechanism, but overall, you can clean the weapon without having to do that. The better designs don't require a screwdriver to dissemble the firing mechanism. Does anyone know of a lever action that does not require a screwdriver to remove the lever? That is a screw, and a screwdriver, that will be lost out in the field. Another important factor is that any military rifle which can be assembled incorrectly, will be assembled incorrectly. (Anyone remember the Ross?) Creating a design which the parts can only be assembled in one order, and it is "intuitive" how they go together, is hard.
There was some great debate between Edward Crossman and Charles Newton, in print, between the lever action and the bolt gun. I have never read any of the discussion, but it would have been interesting to read the arguments these ego manic's created, trying to prove the other wrong.