I could be wrong but I have the impression that Elayne's no 1 loyalty is to Andor with No 2 being the White Tower. She spent most of her life being brough up to think of the interest of Andor first and went to the Tower reasonably late, I think she was around 17/18 years so she had already begun to form her world view.

As far as novice and accepted goes she did not spend very long there as its only two years give or take since the series started and a lot of that time she was out in the West first and then in Tear and back west again. The Aes Sedai have not really had that long to now quite brain wash her but at least indoctronate her into the White Tower world view.

Historically there have been few Aes Sedai who have been queens and sucessful and I can see why? According to the laws and customs of the White Tower Egwene had cause to call Elayne to account for her recent actions but Elayne has a duty to Andor and now Cairhien.

Not to mention how long will it be before some bright spark in the Hall tries to suggest that Elayne as queen of Andor should try to make life difficult for the Black Tower to make them more 'agreeable' to what ever the Aes Sedai want. Elayne as queen must know that for all its power Andor does not want an open conflict with the Black Tower because its Andor who will suffer the casualties and the damange with the White Tower supporting her from the sidelines and out of the firing line.