The Children do so in pursuit of justice against the Shadow. The Aes Sedai do so in pursuit of their own power. The Children at least give sufficient lip service to their ideals that every now and then it is possible for an idealist like Galad to come along and overturn the cynical realpolitik to actually work for that end. The Aes Sedai have no greater ideal, and have transformed the White Tower into an end in itself, so that when an idealist comes along, their equivalent of Galad is merely Egwene, whose ideals are all oriented on the Tower. GALAD has no problem leading a split of the Children when some of the Children have lost their way, but in face of the Aes Sedai being obstructionist and Tower-prioritizing, Egwene cannot or will not split them and lead her faction to the side of those who would battle for their civilization, because she cannot say that Elaida & co are bad Aes Sedai or have lost the Tower's mission. As that mission is nothing more than the aggrandizement of the Tower, their fight is a mere difference of opinion on the way to carry out that aggrandizement. Galad, operating on the stated principles of the Children, says "let's put aside our normal MEANS in order to achieve our ENDS." Egwene, operating on the stated values of the Tower, says "how dare you split for any reason? Most important is our maintaining our normal means, rather than quarrel over the ends for which we have a united Tower."