An interesting fact is that the U.S. had no cavalry until the Civil War period. The earlier mounted force was called Dragoons or Mounted Rifles, and there is a distinction between such forces and true cavalry.
Dragoons rode into the combat area, but then dismounted and fought as infantry. They generally carried heavy pistols and either infantry rifles/muskets or a long arm sized between the infantry rifle and the later cavalry carbine. Sabers, except for officers' swords, were not normally issued.
Cavalry, on the other hand, fought from horseback. That condition meant that a short weapon was needed, and the cavalry carbine came into use. Why a short weapon? Trying to manipulate a full length infantry musket or rifle on horseback would have presented many problems, not the least of which would have been the possibility of knocking an adjacent rider right off his horse with the long barrel.
And fighting from horseback also meant that the saber made sense; backed by both the strength of the cavalryman's arm and the momentum of the heavy horse, the saber packed a lot of punch and (in spite of various comments over the years) really was a formidable weapon.
In the Civil War, some "cavalry" units, especially on the Confederate side, actually fought like the old Dragoons, dismounting and using rifles and shotguns rather than fighting mounted.
In general, though, cavalry feared infantry. The massed fire and longer range weapons of the infantry meant that a cavalry unit simply could not stage a frontal attack against any sizable infantry force. Flank attacks and harassment were the best cavalry could do unless the infantry were either badly outnumbered or very inferior or demoralized. Against a strong and determined infantry, the cavalry would be cut to pieces. (Once, informed of enemy forces ahead, Sheridan asked if it was cavalry or infantry. Told it was cavalry, he responded, "Ride right over them." He could give such an order as head of a 10,000 man unit of three cavalry divisions, a column 16 miles long!)
But the infantry feared cavalry also, to some extent. The constant worry about a surprise cavalry attack led even to some changes in infantry equipment, including the use of a magazine cutoff on rifles, so a full magazine could be kept in reserve in case enemy horsemen suddenly appeared.
Jim