Elaida herself recalled that after her foretelling, which she kept secret, she made sure to attach herself to the girl who seemed most likely to win the Lion Throne.

The evidence is that Elaida "volunteered" herself to Morgase the high seat, and thus got "promoted" as advisor to the Queen of Andor when she won. There was likely no reason for the Hall to refuse to confirm the position, Elaida is capable. Advisors/ambassadors are normally assigned by the Hall (or it's subject to a vote of the Hall anyway, they probably don't initiate all nominations), as far as we know, and it makes sense it's the domain of the "college of Ajahs" as otherwise an Amyrlin would systematically favour people answering to her (it's one of those things Egwene has changed, she's negotiated full power over diplomacy against leaving the Hall final decision over military matters).

It's being demoted as Sitters and being sent as ambassadors to a weak and faraway Court Joline and Teslyn ressented.