Wilder's men were not cavalry, they were "mounted infantry". They weren't expected to fight on horseback, which would have required sabres, they were expected to use their horses simply to get where they needed to go, and then dismount and fight on foot - similar to the concept of the old "dragoons". Because they weren't cavalry, even getting sabres (if they wanted them) would have been difficult because the Quartermasters didn't have a good way to give cavalry supplies to infantry - it probably would have been tied up in red tape until after the war was over. As it was, Wilder had to take out a private loan to get his Spencers, with each man helping to repay the loan with a small deduction from each payroll.
If you read Cozzen's book on Chickamaugua, you will see Wilder's brigade making an oversize impact all over the place - first by forcing the Confederates to abandon Chattanooga when he appeared on the right bank of the river north of Chattanooga (he made himself appear to be a much larger force); later on the north flank of the Union lines on the second day of battle, then on the center, and finally on the flank. In each instance he held his ground and turned back a Confederate advance in his area which threatened to unhinge the Union lines.
Personally, I think he could have caused some problems for the Confederates if he had charged into their flank at the middle of the third day, but I don't think it would have changed the course of the battle. It might have given Thomas more time to reinforce his forces in Snodgrass Hill and the adjoining ridge as some of the Confederate forces were about-faced to deal with a threat from the rear. But the problem which bedeviled all the Union forces on the west side of the LaFayette road was the loss of the wagon trains, which were either captured or fleeing toward Chattanooga. This limited their ammunition supply to little more than what they had on-hand at the beginning of the battle - there were numerous accounts of men being sent in desperate searches for ammunition to supply Thomas' forces at Snodgrass Hill. Ammunition for repeating rifles was particularly difficult to find, one of the regiments on Snodgrass Hill which was using Colt revolving rifles was reduced to using slightly larger rounds, and they found through experience that by attaching their bayoneette to the end of the rifle the bayonette ring provided just enough extra reinforcement to keep the rifles from blowing up in their faces (you have to wonder how many times that happened before they hit upon a solution). Wilder's men might have had more supply than elsewhere - they might have kept their resupply close at hand. But with the number of Confederates involved, I'm sure they would go through that supply a lot faster than they expected.
Bringing the