Haven't denied the utility of cavalry but apart from horse against horse fights can anyone name a battle won by it without missile support or a solid infantry block to retreat to? Belisarius certainly used cavalry effectively in Italy but although the PBI were despised by writers, on his campaigns it does seem as though he relied on his foot.
Keegan makes the point that a cavalry charge against cavalry is nothing like a head-on clash but the two units would ride through each others group. In effect whatever the size of the two groups it was always like a skirmish. Lancers might have behaved slightly differently because their weapon required a point of aim on an opponent but even so it was never a solid front that struck their opponents.
Is it true that Custer's men lacked sabres and if so did they have pistols with which to skirmish? It seems to me that they tried to act like mounted infantry, dismounting and taking cover behind their horses. Had they been trained in controlled volley fire? The Indians had been called the finest light cavalry in the world but is that only in comparison to their opponents?
I'll try to cover some of your points one at a time.
Starting with your last question, Custers men did have sabers. BUT! From the civil War on sabers were obsolete. According to diaries from the troops, if any cavalry unit (even during the war) attacked with drawn sabers the other side "paid them no more mind than if they had been armed with cornstalks." It's true however that Custer's men fought more like mounted infantry. NOT because of their use of pistols (a mounted charge with pistols is still a cavalry charge) Rather because they dismounted and fought from defensive positions with rifles. And yes that was more common out West after the Civil War.
Your first question asks about successful cavalry charges without infantry back-up:
1. The one I posted regarding the 26th Cavalry against the Japanese in 1942 in Morong
2. The Australian Lighthorse in Beersheba in WWI
The switch from sabers to pistols doesn't change the fact that they were cavalry any more than the switch TOO sabers from lances did. As long as they stay mounted and can perform an effective charge it's still a cavalry tactic.