Throughout the series, and specifically when she believes her mother dead, Elayne takes for granted that she will assume the throne of Andor.
Yet when Rhavin conquered Andor from within, Elayne was not present to fight the threat. Rand was.
When Andor faced anarchy and poverty after Morgase and Rhavin were both gone, Elayne did not govern for the benefit of her people. Rand did.
These prior two points can potentially be laid to the fact that Elayne-as-Accepted did not possess freedom to return to Andor. However, it's pretty clear that Nynaeve, Egwene, and Elayne pretty much do whatever they wish and go wherever they want after book 3.
But even if that argument did hold water...
The minute Elayne was declared Aes Sedai by Egwene, she did not return to Andor. She went the complete opposite direction, to Ebou Dar.
When Elayne reached Caemlyn, she did not publicly denounce the rumors of Rand killing Morgase.
In fact, so grateful was she for Rand's help in defeating the Forsaken ruling Andor, and keeping Andor from the civil war that tore other nations apart that she took down all of his banners.
Of course, one could claim that Elayne was ignorant of these events, but had she behaved like an heir should and returned to Caemlyn at the first hint of trouble, she would know all of it.
Instead, she shows up johnny-come-lately to assume the throne, taking advantage of Rand's goodwill towards her without so much as a thank you. She doesn't even acknowledge that if the Dragon Reborn WANTED, he COULD conquer Andor which is extremely weak at this point, and only Rand's goodwill keeps the Aiel, other armies, and Asha'man from treating Andor like a rebellious nation which hasn't acknowledged the Dragon Reborn.