Without the White Tower struggle a whole bunch of things might not happen. The White Tower became engulfed with its own issues it couldn't interfer with the rest of the world.

1. Rand would not have been able to accomplish what he needed to do. The White Tower would have tried to "guide" Rand, into what the White Tower thought was necessary. After Rand conquors Cairhein would he have been meet with 30, 40, 50 sisters who have been available to guide Rand. With the sisters guidance would Rand have been able to attack Rahvin? With the sisters guidance would Illian be part of Rand's empire? With the sisters guidance would there be a black tower?

2. Would Elayne be on the Andor throne? Elayne was an accepted iniate of the Tower. Elayne may have been allowed to go to Andor and earn the throne (with White Tower guidance), but she would have been forced to return to the Tower to finish her training. The White Tower would have been guiding whoever was named regent. Would the White Tower "advisor" advise the regent to support Rand's goals or the White Tower goals? Andor is now firmly allied with Rand.

3. The Bowl of Winds, angreal, and ter'angreal would not have been found be the Super Girls. The White Tower would have scooped up Nyneave and Elayne and returned them to the Tower. The weather would not have been righted causing millions of deaths. I angreal and ter'angreal will be important. Both Elayne and Aviendha have angreal, the bone handled knife Elayne has will hide her from the Dark, the communication and library ter'angreal will all be important.

4. Elayne, Aviendha, and Min would not be bonded with Rand. The benefits of that bond are becoming apparent. As an example if Rand tranmigrates into Moridin's body the bonded girls would know who Rand was.

5. Egwene would not be Amyrlin (I know Cannoli you are having an apoplectic fit). With Egwene as Amyrlin the White Tower will think about helping Rand instead of leading Rand.

6. Without Elaida in charge 50 sisters would not have been sent to the Black Tower and ended up getting bonded. As a result 50 sisters will bond 50 Ashaman. Rand not has at least 100 teams of Ashaman/Aes Sedai who are bonded and could link in the final battle.