In 1890, the United States Chief of Ordnance remarked:
Several European nations, during the past few years, have made premature changes of caliber or have adopted crude repeating systems that have had to be abandoned for newer and better ones, often before the armies were fully armed with them.
While this statement was about the need for patience in testing a magazine gun for the army on the eve of the 20th century, it may as well have called the Dreyse needle Gun out by name. The Needle may have been one of the first standard issue breach-loaders in use by any European army, but that didn't mean it didn't have its problems. It fired "caseless" ammunition - a soft cloth, treated cartridge with an internal primer - had difficulty with sealing the breach, with cleanliness and reliability, et al. The Needle gun was outclassed by other competitors quite soon after its adoption.
You would find similar opinions about the need for patience expressed in the 1870s and the 1860s and 1850s. Innovation was great, and rate of fire was great, but not at the expense of a functional, reliable firearm. This wasn't only the opinion of the United States Army, either, most European countries in the 1860s were still equipped with muzzle-loading rifles, though several countries (including the United States) were developing military-facing breech-loading rifles.
While the United States had done some testing with repeaters, it hadn't adopted one for standard issue for quite a few reasons: the Ordnance Board testing platform stressed reliability and ease of use more than rate of fire, owing partially to the perception quoted above - new stuff breaks more than old, proven stuff, and it would be a waste of time and money and possibly lives to rush ahead to adopt the cool new thing when the old thing works; the US Army before the Civil War was tiny and it budget was far more limited than it is today, the function of any rifle adopted as standard must also be understood to be something that can handle the rapid expansion of the army in times of need, which means that any standard platform should be capable of rapid mass manufacturing; breech-loading in the 1860s had been around in some cases for decades, it's true, but the technological bottleneck wasn't a closing breech block, it was ammunition; the US Army had tested a number of breech loaders and had a few different versions in field tests throughout the 1850s; and finally, the war interrupted the regular testing program, and the army shifted from testing with an eye to quipping the entire army to field-testing known rifles in small batches. Volunteer regiments also regularly equipped themselves with different rifles.
Let's go through these one by one. I've written a lot about the US Army's testing program. As a quick summary, the army was looking for a rifle that would serve for all of its soldiers without the need for extensive and expensive field maintenance, one that didn't break often, one that could be manufactured quickly, and one that was popular among its officers. During the tests after the Civil War, the Ordnance Board enthusiastically recommended the "trapdoor" system in part because field officers in field trials overwhelmingly preferred it to any other options - out of 95 field officers given rifles for trial, 84 preferred the trapdoor. By the time he Civil War had broken out, there was no such front-running favorite, even though a couple of rifle models had been ordered for field tests.
Second, the army's small size and the political resistance to spending a lot of money on it was a political feature of the United States prior to the Cold War that shouldn't be underestimated. There were 16,000 men in the US Army before the Civil War. The Union army soon ballooned to orders of magnitude more than that, and the simplicity of the muzzle-loading model 1855 rifle-musket allowed domestic manufactories to produce 1,700,000 of them during the war, while also producing more than 300,000 breech-loading rifles and carbines. The US also imported more than a million rifles at the same time, of various makes and models. While volunteer regiments had a lot of leeway to equip their men as they saw fit, those rifles still had to come from somewhere, and domestically manufactured rifles were sold to volunteer regiments and imported for volunteer regiments; all of them had to come from somewhere.
Third, ammunition. The army had only in 1855 settled on a version of the Minie ball that they liked, and the big issue with a lot of breach-loaders was that they were still being designed to fire "caseless" ammunition. That is, either (like in the case of the Jenks carbine) with a ball and loose powder, or with a cloth cartridge (early model Sharps, and even the Needle gun, as above). Experimentation with cartridges - another complex testing program by the army Ordnance Board - was ongoing in the 1840s and 50s, but no satisfactory single metallic cartridge had been chosen by the beginning of the war. By its end, quite a few metallic cartridges had seen hard field use and made their way into post-war testing programs, which eventually produced the "modern" metallic case cartridges we're familiar with today.
Despite this, the army had at least two ongoing field trials at the start of the war, with the Jenks "mule ear" carbine and the Sharps. The Jenks was very odd but extremely rugged and dependable rifle, which had earned some fame in wtihstanding more than 14,000 shots in an endurance test before the nipple - where percussion caps were loaded - broke and ended the test. Small numbers of Jenks carbines were ordered for the navy.
Most of the breech-loaders available to volunteer regiments and regulars at the start of the war were firing "caseless" shots, loading either cloth cartridges or loose powder, and firing them off with percussion caps. By the end of the war, several viable options for metallic cartridges had been introduced, including rimfire and centerfire "self-priming" options.
To make a long story short, the United States hadn't fallen behind anything, they were still regularly testing various viable options for reliable breech-loading systems to adopt as a standard arm. This was totally in parallel with most European armies. The French Chassepot was only adopted in 1866, and it took until 1871 for the British to adopt a purposefully-designed breech-loader in the Martini-Henry, which replaced their conversion breech-loader Snyder-Enfield.
What I think is a valuable lesson to take away from all of this is that looking at firearm development from a modern perspective tends to dismiss or distort the on-the-ground reasoning used by contemporary soldiers and designers, and look for the thing that looks the most like we think firearms should look like; early breech-loaders are, in this way, viewed as "advanced" which makes the alternatives look "obsolete" while ignoring the problems pointed out by the ordnance board in the quote above. Rapidity of fire was not the game-changer we like to think it was. It was not a hallmark of a superior rifle, it was just one concern in a complex equation that had to balance rugged use, manufacturing, dependability, and dozens of other factors.
David F. Butler, United States Firearms: The First Century: 1776-1875
Gregory Urwin, The United States Infantry: An Illustrated History 1775-1918
The U.S. Army in the pre-war era, say 1860, looked towards the French military for it's doctrine, and France wasn't using breechloaders in 1860, they were using rifled muskets. America was not alone in being "behind" the Prussians and Scandinavian armies that adopted breechloaders early, ALL the big prestigious armies of Europe: French, Austrian, Russian, and British were behind the Prussians.
America in the 1850s had a tiny military with few professional officers compared to European states. The country made a political choice to forego a standing army and rely on volunteers and militia for national defense. Accordingly, American military officers knew they would not be at the cutting edge of military doctrine. Therefore they looked to Europe for models for the best way to organize and equip the army.
France seemed to Americans to be at the forefront of military affairs. It had beaten the Russians in the Crimean war in 1853-6. France had defeated Austria in 1859 at the battle of Solferino. The French colonial troops, called Zouaves, were widely considered to be some of the best troops in Europe for light infantry tactics, and had much combat experience in North Africa. (Plus their uniforms were soooo fashionable.) In contrast, the Nordic countries equipped with breechloaders had not fought a war since a German civil war in 1848-49.
American officers who later became prominent in the Civil War like Hardee (famous for writing a tactical manual) and McClellan went to Europe to observe the French army and French equipment. They convinced the American military to adopt french style arms like Minié ball rifled-muskets and smoothbore 12 lb "Napoleon" cannon, which were named after the sitting French emperor, Napoleon III not the earlier emperor. Aside from shortages, (big aside) American equipment was pretty much up to the European standard.
But the Europeon standard moved rapidly after the American Civil War. In 1866, the Prussian infantry crushed the Austrian troops at the Battle of Königgrätz and it became immediately clear to European military establishments that the Prussian use of breechloaders gave them a substantial advantage. After that, European armies adopted the breechloader and began a period of fast moving technical advances that dramatically increased firepower and transformed tactics for all branches of the army.