Logistics were generally at the core of the issue. The Civil War saw the Union and Confederacy raise armies larger than the US had ever seen, and as they did so, supply became a bigger concern than the individual effectiveness of an infantryman, and repeating rifles available at the time were significantly more expensive to procure and use.

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For some context, it helps to understand the standard weapons of the day. The standard infantry rifled musket was a muzzle-loading rifled weapon that had some major improvements over the Napoleonic-era muskets they're usually compared to. Percussion caps had replaced flintlocks, which simplified the reloading process for soldiers, and the introduction of the Minie ball finally provided a means for rifled muskets to be practically used by line infantry. Troops would generally carry 40 rounds of ammunition, with 60 rounds being a "heavy" load that you'd see in situations like a force slated to be at the front line of a major assault on fortifications. The ammunition itself was generally carried in paper cartridges that contained ball and powder, again simplifying the reloading process. With these weapons, you'd still have glacially slow reloading speeds by modern standards (~3 round per minute) despite the improvement over older flintlocks, but the rifling would massively increase their effective range compared to smoothbore weapons. Rifled muskets of the time generally had adjustable sights going out to several hundred yards, and the British Pattern 1853 rifled musket had sights that reached out to 1,250 yards. Although engaging at extreme ranges may have been rare, the increased effective range of the rifled musket was an important element of infantry doctrine of the time.

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Compare that to the repeating weapons of the day. Repeating arms were by nature more complicated and expensive to produce. American arsenals at the time were tooled up for mass production of rifled muskets, and even as they expanded during the war they struggled to keep up with demand. The myriad of repeating arms that saw service in the war were private ventures of various arms makers, and often times these manufacturers were struggling to keep up with the comparatively small orders they did receive, let alone a whole army's worth of rifles. The special cartridge ammunition used by rifles like the Spencer complicated things even more, as it production wasn't something that could be done easily by existing ammunition manufacturing facilities. Cartridge cases needed to be pressed from copper or brass, priming material needed to be put in the casing, and powder and bullet would need to be put in the case - something far more intensive than rolling up a bullet and some powder in paper.

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Doctrine then became an issue, as there wasn't yet any established practices for the use of repeating arms. There was a fear of panicked soldiers expending all their ammunition (a very real problem in a war without consistent access to secure railheads) and leaving the army almost defenseless, and reloading became a major concern, as these early breech-loaders generally had very time-consuming reloading processes. Certain repeating carbines also "suffered" from lacking certain features that would be necessary for a standard infantry arm - namely, infantry-length barrels (~30 inches), the ability to mount bayonets, and a cartridge capable of reaching out reliably to the distances armies expected to engage at. That was a liability for guns like the Henry repeater, which was chambered in a cartridge that had a very short effective range. These issues and concerns would actually outlive muzzle-loading arms - fears over supply being impacted by rapid-firing guns didn't really go away until WW1.

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All these issues aside, we do see repeating arms used in increasing numbers as the war goes on, particularly in the Union. Cavalry forces were more regularly equipped with repeating arms, with Sheridan's troopers in particular standing out in the 1864 campaigns for their heavy use of repeating arms. I can't speak authoritatively as to why cavalry tended to get the weapons first, although cavalry historically has had a legacy of being armed with smaller, lighter weapons with shorter effective ranges, so many of the concerns about the guns themselves listed above weren't as pressing. Infantry units would also occasionally be outfitted with repeating arms, and often they performed spectacularly well. One unit at Chickamauga was armed with Colt Revolving Rifles and managed to hold its ground despite the army nearly collapsing around them, although they did nearly run out of ammunition during the battle. During Grant's Overland campaign in 1864, one unit in the Army of the Potomac also was equipped with some form of repeating arm, making an impression on Confederate troops they faced. However, repeating arms for line infantry were generally nonstandard weapons and thus their procurement would be something done by private initiative, often from a commanding officer.