Gawyn is pre-whipped. All political power is based on force of arms, and in a world where those are all muscle-driven a female ruler can easily lose the respect of her soldiers by her inability to fight. Plus, Andoran noblewomen seem to almost take pride in their lack of military prowess and leaving such things to the men. That could lead to a popular general like Gareth Bryne or someone similar leading a coup. The Andoran answer to this problem is to make the Queen's eldest brother the commander in chief over and above any officer who was promoted on merit, thus, there will always be a top military commander with blood-ties to the Queen. However, in order to prevent such a Prince from taking the throne himself (along with the supposition of family loyalty comes a blood claim on the throne), he has to be trained from childhood to be a faithful and obedient subject. It is not enough for him to be loyal, as loyalty could manifest in opposition, in taking her down for the good of the realm, the First Prince of the Sword must ultimately be obedient. As we see from the very first book in the series, if Gawyn has a fault on that scale, it is to be too obedient, rather than not enough. Morgase reprimands him for not being more of a check on his sister, because he blindly goes along with whatever she orders, no matter how much he protests.

Egwene gets a man who has been conditioned all his life to obey a woman her age. And he's still a little too independent for her taste!