Let us, for a moment, ignore your perception of Egwene and her plans for the White Tower.

Think of this from Elayne's perspective: She is a queen of two nations, mother to the children of the Dragon Reborn, technically the liege to Perrin Aybara and the Two Rivers, currently the employer of the Band of the Red hand and its canons, currently the only ruler to have Channelers around her to aid her in Healing and trade. And she is one of the most powerful Aes Sedai in the world, the only maker of ter'angreal (thus making the Tower beholden to her), extremely skilled in working the weather, and short of the Amyrlin Seat, the most powerful woman in the White Tower organization, and likely to be seen as her equal in influence and power.

So she has two options. Make a needless fight with a woman whom she respects and who respects her, with whom she has shared a multitude of dramatic experiences, who would make a promise to treat the Windfinders the same as Wise One's based on just her word, who offers her soldiers to win her throne even when she herself is soon to enter a difficult siege, who has clearly shown herself to be adept at playing politics and who has shown no care that Elayne usurped her authority and burned the rings of the Black Ajah sisters...

Or, she could use the close relationship she has with this woman to their mutual advantage. Together, they form the single most powerful entity in the world, be it in therms of number of channelers or wealth or armies or political influence. She is in the innermost councils of a woman who apparently can successfully call the rulers of the world to her with their full armies even when they are sworn to the Dragon Reborn. She has already been able to influence this woman in powerful ways (Windfinders, the Bowl of the Winds...). While she herself can also be influenced by Egwene... why should she care? All evidence to date has shown that they have only benefited from cooperating. The Tower and Andor both are today much more stable than they have been in a long time. Rather than dealing with the Tower and the Amyrlin Seat as unreachable entities as most rulers mush, she is instead part of a small council of women who are truly running the Tower. She's be an absolute moron to want to change that.