Political alignment is not the same as shared benefit, and the presence of an Aes Sedai on the throne of Andor may paradoxically make the Tower weaker there than it might be elsewhere. Most rulers have an Aes Sedai advisor, even if not openly, and listen when she speaks. It's a kind of insurance, and the Tower can discreely blacklist those who don't play ball, cutting them out of the best diplomatic and trade deals. This has been their modus operandi for thousands of years. But Elayne is an Aes Sedai herself, and her shawl gives her all the privileges of association with the Tower with none of the drawbacks of having to do as the Hall of the Tower thinks best.

This would not have been true, I don't think, of Moiraine, if they'd managed to make her queen of Cairhien. Her NS POVs make it clear she was well indoctrinated. But Elayne spent very little time in the Tower. She's still capable of independent thought.

The presence of the Black Tower may change matters, though. Assuming it isn't entirely destroyed, and assuming it and the White Tower remain separate organisations, I wouldn't be surprised to see it slowly take over Andor, or other countries, in the coming decades and centuries. Certainly it's already a legitimate political player, purely by virtue of having the power to reduce Caemlyn to molten slag if it wishes.