Egwene has been very open in stating that she means to take the White Tower to new strengths. She isn't even remotely secretive about it, and I fail to see why she should be.

She worked out a deal with the Wise Ones and the Windfinders to tie them all together. That, in itself, both strengthens the Tower more than it has ever been, but also makes it impossible for the Shadow or the Seanchan to destroy one of these organizations easily.

It is the same with Cairhein. In a time of war, it makes much more sense for the resources of Cairhein and Andor to be used in concert. Since Tar Valon is strategically located roughly in between these two nations, and will undoubtedly be a focal point for resistance against the Shadow or the Seanchan, it makes excellent sense to have a unified command to tie these three together.

In one stroke, one of Egwene's closest friends and allies becomes the most powerful person in the WT excepting Egwene herself. As a powerful Aes Sedai who rules two kingdoms, Elayne will be a powerful ally for Egwene if the Hall should try something stupid. Of course Egwene wants her to have Cairhein. Why wouldn't she?