Direct comment: There has been reference made to Custer being able to select repeating rifles. This shows some ignorance of the state of Ordinance in the Victorian era Army; riflery and marksmanship were given no real attention until the 1880s and the situation was such in the Custer era that the targets were made oblong to account for inconsistent long range trajectory of issue ammunition rather than improve the ammunition. Col. Custer's sole choice was what the Ordinance Corps authorized: The Model 1873 Springfield Carbine and the .45 calibre cartridge backed with 50 grains of black powder.
The carbine load at that time used the same 405 grain bullet propelled by a reduced charge of 55 grains of black powder. To take up the extra space a cardboard tube was inserted in the case with the powder then filling the tube under the bullet. At LBH .45-70 cases have been found that still contain the cardboard tube.
In the wake of Little Big Horn, many people were critical of the board that adopted the 1873 Springfield carbine,when repeating carbines were available - but a single shot carbine fit the doctrine of the time.
It was felt that troops would ben wasteful with a repeater - and that attitude didn't die until WWI. You'll note the magazine cut off on the 1903 Springfield - it's there because the doctrine of that time, almost 30 years post Little Big Horn, was that troops should single load rounds and keep the five rounds in the magazine in reserve for the 1903 equivalent of "final protective fire" that was applied to the use of fuel automatic fire in the M16A1 70 years later.
The longer ranged 45-70 also fit the tactical doctrine of the day better, and at the time the repeating carbines available all shot pistol caliber rounds that provided a maximum effective range of around 125 yards in black powder form. In contrast the Sharps Calvary carbine was effective well beyond 200 yards.
The war department also wanted to standardize ammunition and since the .45-70 was the standard infantry rifle there was a substantial incentive for a 45-70 cavalry carbine (and not all that successful given they quickly adopted the lighter carbine load for it).
Training was a major issue at Little Big Horn. A that time there was no basic training - all training was done at the unit level and it was system that didn't work all that well. At LBH, the cavalry troopers almost to the man fought as dismounted infantry and that to a great extent reflects their limited training. Unfortunately, at LBH it also meant they sacrificed the mobility that could have helped them maintain the initiative and disengage successfully.
Marksmanship training took a back seat to conservation of ammunition and as I recall the amount allotted per trooper per year for practice was on the order of 20 rounds. It's likely that more shots were fired by troopers for hunting purposes to put fresh meat on the table, but the total amount of live fire training was woefully inadequate and did not include such advanced skills as firing from horse back, thus reducing the cavalry to mounted infantry using horses to get to the scene of the fight and then dismounting.