For a few different reasons. First and foremost the lever action rifles were still very new at the outbreak of the war, the first production ready design was the spencer repeating rifle which was not ready for production until 1860, just a year before the war. The Spencer rifle was also many times more expensive than a musket owing to it's relatively complex design and small production numbers which absolutely factored into the equation. The high rate of fire was not just a positive thing. While a lever action rifle can achieve rates of fire 10 times higher than a musket, thus providing a tactical advantage, there are drawbacks to that. High rates of fire consume much more ammunition and generate much more smoke, both of which were already a problem in the civil war. Finally the introduction of lever action rifles was opposed by parts of the Deperatment of War that felt that introduction of rapid firing rifles would lead to soldiers wasting too much ammo.

That said, lever action rifles, specifically the Spencer rifle and carbine, did see action during the civil war and were quite popular with the men who used them. A few notable battles that saw use of lever action rifles include Hoover's Gap, Hanover and East Cavalry Field (the latter two under Custer).

TK;DR - They were more expensive, consume much more ammunition and the Department of War were reluctant to introduce them in the midst of the war due to supply concerns.

There's a variety of reasons, many of them relating to logistics, although the standard rifled muskets in use at the time did have some niche advantages over the lever-guns available at the time.

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Most obviously, there's cost and availability - an issue that extends to even the breech-loading designs of the day as well as the few magazine rifles available at the time. Compared to a rifled musket, breech-loaders and magazine rifles were more expensive and complicated to produce. Compare the common rifled muskets being procured during the war - the Springfield 1861, Lorenz rifle, and Enfield Pattern 53 all around $20 per gun - to repeaters like the Henry or Spencer repeaters (around $40 per gun) or even comparatively simple single-shot breechloaders like the Sharps ($30 per gun). These guns were significantly more expensive than what was already in service, which was a major liability when finances were already being strained by the massive expense of the war. These breech-loading designs also suffered from the fact that they were privately developed and owned, and, unlike the rifled muskets built in state arsenals, they lacked the tooling or production capacity to provide the kind of output the military would need.

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Along the same lines, there's also ammunition. Repeaters like the Henry and Spencer used rimfire ammunition as opposed to the paper cartridges of the Sharps and rifled muskets. While brass-cased ammunition may be the standard today, paper cartridges offered logistical advantages for armies of the time. Ammunition could very easily be put together in the field - lead could be cast into bullets, powder pulled from stores, and paper used to divide up into convenient packages. Rimfire ammunition, on the other hand, had to be produced in a factory that could produce the brass cases. Rimfire ammunition also suffered from being a fairly new technology, really only taking shape in the 1850s and not being fully established by the time war broke out. Issues exist with rimfire ammunition to this day due to the method of putting the primer in the rims, and reliability is often spotty.

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Ammunition would provide the most glaring performance shortfall of the Spencer and Henry as well. Compared to the standard rifled muskets of the time, which fired a ~500 grain (~32 gram) projectile at about 1,200 ft/s, .56-56 Spencer was only a 350 grain bullet at 1,200 ft/s and .44 Henry was a 200 grain bullet at about 1,100 ft/s. While the rifled musket ball had energy comparable to an intermediate rifle caliber of today, .56-56 falls well short of that, and .44 Henry is closer to the energy of a pistol cartridge. Rifled muskets of the time often had sights out to several hundred yards, with the Pattern 53's sights going all the way out to 1,000 yards, with the idea that volley fire could be used at longer ranges with reasonable effectiveness over an area. The repeaters of the day lacked the energy to reach such ranges, and thus were incompatible with this doctrine. The Henry in particular had major issues, with its effective range by modern standards being only around 100 yards (although soldiers likely pushed it further).

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And then we have doctrine. Really until after WW1, many armies were hesitant to adopt magazine rifles due to concerns over ammunition expenditure. Commanders feared that a magazine rifle would allow soldiers to dump ammunition in a panicked and undisciplined manner, running out of ammunition with little effect. Even when magazine rifles were finally adopted as a standard in the 1890s, the US was among several nations to introduce magazine cutoffs on their guns (for the US, both the first-generation Krag Jorgensen and the significantly more modern Springfield 1903 that replaced it) specifically due to these very same fears over magazine rifles. During the Civil War, this was a legitimate concern due to the relatively poorly developed rail network in the South and the nature of the campaigning, and we'd see this kind of thinking carry on in the US in part thanks to the very sparsely populated American West where a lot of future fighting would happen.