In otl the Battle of the Crater seems to have been a lost opportunity. Suppose competent logistics had been planned and the USCT (United States Colored Troops)been able to take Petersberg, and maybe reach Richmond by September 1864 how different might things have been?
Notably how likely is it that Lincoln would have been more inclined to press for voting rights for former slaves?
If the Union wins at at the Crater, Petersburg falls, and then Richmond falls in a matter of days; no later than mid-August. USCT may lead the march on Richmond, or may not. It won't be the same troops as fought at Petersburg. OTL, it was the troops posted north of the Appomattox (and James) that occupied Richmond, while the troops around Petersburg marched west after Lee.
However - if the Crater attack had been carried out as originally planned, by the originally assigned division of USCT - it could have succeeded. Richmond would fall and the war would end by fall 1865. (It would take longer, because Hood's Army of Tennessee was still effective.)
Lincoln rolls to easy re-election. Then what? Lincolnian Reconstruction follows. Lincoln would raise the issue of votes for blacks, as he started to do OTL. Would the prestige of the Crater victory prompt him to do so earlier and more vigorously? I would like to think so, but I cannot say for sure.
I would note that the issue was not "votes for former slaves", but "votes for blacks". Many of the USCT were free blacks from the North. There was strong anti-black prejudice in much of the North, including opposition to black voting, but it was not a threat to the established social order as it was in the South. If USCT veterans in northern states demanded the franchise, they could probably get it, at least in Massachusetts. That might set a precedent which Lincoln would want extended elsewhere, even the South.