It sounds like you're generally referring to what's known in the era as Light Infantry tactics. Cover was used when available, and in many places light infantry tactics absolutely were relied upon. There's plenty of other posts out there that have explained the rationale of the Napoleonic line infantry tactics (mainly a defense against cavalry and better command and control), and most of those reasons generally held in the Civil War as well.

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That's not to say cover wasn't used quite extensively and effectively. The Sunken Road at Antietam was one of the more famous cases where improvised breastworks were combined with a low-lying road to provide devastatingly effective cover for defenders, and the battles of Fredericksburg and Corinth both saw the successful defense of prepared positions by the Confederacy and Union, respectively. By the end of the war, you had both sides erecting field fortifications fairly regularly. However, the technology of the time still had to be taken into account with such defensive positions.

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The standard infantry arms of the time were muzzle-loading rifles. While more modern than the flintlocks of old (they relied on percussion caps to ignite the powder), they still had a reloading process that, by the standards of more modern breechloaders, is fairly lengthy and cumbersome. Going prone makes this process even more difficult than it already is - these are long rifles and you need to be manipulating them to ram a powder and ball down the barrel for every shot. For skirmishers and similar forces whose role isn't to be the main line of battle, making more use of cover, including going prone, can be worth the tradeoffs with respect to things like reloading and command and control. For the main line of soldiers, however, going prone impacts the ability of the unit to make use of its multiple ranks (either massed fire by multiple ranks at once or fire-by-rank), limits their mobility, and leaves them vulnerable to artillery and cavalry. Effectively you'd be limiting density of fire in a way that would reduce the effectiveness of the unit and leave them vulnerable to being overrun in a charge.

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One thing to note is that the tactics of the time weren't exactly perfect. While at a glance the Civil War seems little different in terms of technology from the Napoleonic Wars, advances in weapons technology meant that a lot of the tactics of old were rapidly becoming untenable. Rifled muskets and more advanced artillery were extending the ranges armies were engaging at and increasing the effectiveness of infantry and artillery fire to the point that cavalry charges were much less of a threat than they once were. The more heavily wooded terrain of many battlefields often helped to reduce the distances forces had to charge across open ground, but instances like Pickett's Charge at Gettysburg show just how devastating the new technology could be - Pickett's division was shattered with around 6,000 casualties from charging across only a few hundred yards of open ground.