The drums definitely have lot to do with it but this is probably the bigger reason. Listen to the New York '99 bootleg for example. Aces High is a bit, uh, up there and Bruce is definitely having a bit of a time trying to keep up physically getting the words out of his mouth at more or less the correct pitch:
Note that the tempo goes up by a fucking lot on this take, like it does even on the album version. The end harmony sounds like a different song compared to the one in the beginning. I might even argue that this is what it should sound like but understandably the 90s were probably the last time it was feasible (if barely) for them to play this fast live.