The late gun writer Elmer Keith (1899 - 1984) began carrying a Colt 1851 Navy in 1912, when he was 13.
He grew up around Civil War veterans in western Montana. Two of the veterans, one a Confederate and the other a Union soldier, saw a great deal of action.
Keith wrote that both veterans told him that the conical bullets were good for penetration, but a ball over a full chamber of powder had much more stopping power.
Keith agrees with this, citing his experiences in shooting game with conicals or balls.
"The conical pointed bullets gave more range and penetration than round balls, but never were as accurate in our guns, nor did they kill game as well as the round ball. The pointed bullets seemingly slipped through game with a very small wound channel while the blunt round ball at fairly high velocity, tore a good wound channel all the way," Keith wrote in his book, Sixguns released in 1955 (and I believe still in print).
That said, archaeological surveys of Civil War campsites and battlefields shows that nearly all projectiles used in revolvers were conical. The concal bullet, with its attached paper tube containing gunpowder, was standard issue for both armies.
If you wanted balls, you likely had to cast your own, and I don't see many soldiers having the time or resources to do that. Ever try casting bullets in a campfire? I have, and it's not as easy as depicted.
Very few pistol balls have been found where the armies camped or fought.
