It is something of an apples-for-oranges question. Though it had a slower rate of fire, a rifled musket had a longer range, better accuracy and more power than a revolver. It could also take a bayonet.

The War Dept. would actually buy around 4,000 Colt New Model Revolving Rifles, starting in the mid 1850's, that had more power than the pistols, and more accuracy. However, these were just one of several breech-loading weapons that were issued, and they were almost all issued to cavalry. It was recognized that loading a muzzle-loading gun on horseback was awkward, even a short carbine buckled to a sling.

The War Dept was aware that muzzle-loaders were more and more obsolete- the recent wars in Europe had shown that. But there was , above all, the need to be able to equip an entire army. In contrast to the hundreds thousands of breech-loading guns purchased from various manufacturers for the war, around a million of the 1861 Model Springfield Rifled Musket alone were made, 700,000 more of the Model 1863, and hundreds of thousands more rifled muskets imported. There was also the enormous challenge of supply: the great variety of ammunition needed for the great variety of breech-loading guns gave the War Dept. fits, after it had to produce millions of cartridges for the .58 muskets it already had.

A myriad of reasons, but the primary one is effective range. The most common muskets were firing a 500gr projectile with around 65gr of powder behind it out of a long barrel, whereas the largest common revolvers were firing a 140gr projectile with around 30gr of powder behind it out of a much shorter barrel. That's a big disparity in weight and powder charge, exacerbated by the difference in barrel length (longer barrel gives more time for the powder to burn and impart energy into the projectile) all of which ultimately means that a musket has an effective range at least 4 times greater than the pistol. In practice this difference is even greater, because aiming pistols is far more difficult. In the hands of a skilled shooter you can range a pistol out to 40-50m at most, but that takes years of practice, and in the hands of a conscript the effective range drops to closer to 15m. Conversely, it is fairly easy to train someone to shoot a rifle effectively out to 100-200m

All this to say, if I can shoot you from a distance 4-5X further than you can shoot back, then it doesn't matter how quickly you can shoot, because you can't hit me anyways.

Additionally, it's important to consider that these are still fundamentally muzzle loading firearms. Yes, you have a 5-6 round capacity, but once that's expended, you still need to load each of those rounds individually. This is the reason early cartridge repeaters often came with magazine cutoffs, allowing for soldiers to load and fire single rounds, but have a reserve capacity for emergency. For sustained fire, assuming it takes say, 20 seconds to load. Firing once, loading for 20 seconds, then firing again, is effectively the same as firing 5 times, loading for 100 seconds, then firing 5 more times. The first set of 5 is faster, but if you sustain fire they approach the same effective rate of fire relatively quickly.

Finally, there is the always relevant factor of cost. There were, for example, breech loading rifles that were pretty clearly superior in both range and rate of fire, but generally speaking militaries are reticent to adopt new standard weapons during times of war. At the time war broke out, they already had the ability to make hundreds of thousands of muskets, and they needed to arm hundreds of thousands of soldiers immediately.