Assume for a moment that once Siuan and Leanne were removed, no one left the Tower and Elaida had complete mastery over the Aes Sedai.

Several things would have changed:

1) The disastrous attempt to kidnap Rand would not have ended where it did. Elaida would have mounted a rescue effort to free the Aes Sedai. Only the rebellion stopped her from doing more.

2) Same with the Black Tower expedition.

With just these two events, both Rand and the Tower would have been weaker at the other's expense. The BT never would have become a big thing, and Rand would never have been able to achieve as much as he did.

Also, Nynaeve, Egwene and Elayne would not have been Aes Sedai. This means no Bowl of the Winds, which means the world would have been in severe drought at this time. This also means that there could have been interference with the Cleansing.

Also, Elaida may not have taken to the upper floors. Who knows, if she had been seen as Amyrlin by everyone, she may have gotten her palace completed faster, and the Seanchan would never have taken her.

But the primary purpose of the Rebellion, as Egwene and Siuan saw it at least, was to prevent Elaida from bungling with Rand. Egwene even states in tPoD that this is why she allows Siuan to lie about the Reds!

And they achieved this purpose. Rand was able to do so much because a unified Tower under a Red was not present to oppose his every move.