The original story was a coming of age story however rafe of time aged up the characters and basically got rid of any innocence they have so it certainly isn't anymore.
Also the white tower doesn't hold supremacy it has a lot of power and influence but there are countries that are not under their sway and those that pay lip service as well. If the white tower held supremacy many of the events that happen in the books couldn't happen because it would be aes sedai puppets being displaced rather than rulers which would bring the full weight of the tower down on the replacer.