I think you mistake not having subjects bound to obey her every whim without debate with not having a power base; as the lawfully Raised Amyrlin, all initiates of the White Tower are loyal to her, and it would take a lot of mistakes on her part to lose it. It may not appear this way to us because we have privileged information, but there was a reason that openly opposing an Amyrlin was viewed with shock - this is a deeply democratic institution. The fact that there is discussion, open debate and internal structures that balance the power of the Amyrlin shows its' health as an organization. Many of the changes she has implemented has made the White Tower more healthy than it had been before - especially some of the secondary outcomes of her actions (ie: the law that any new course of action needed the full Hall to review before approving it means political backstabbing and secrecy will be reduced).

Egwene is a scholarly type with a drive to best the best at whatever she sets her mind to. I believe that you confuse this drive with a lack of identity. She didn't feel a need to "be the best damane she could be!" once captured by the Seanchan, because this drive is very much connected to her sense of independence and self-control. In other words, she felt empowered by the Aes Sedai and the Wise Ones, and hastened to learn as much as she could from them. In my opinion, this is a good quality for a leader to have (especially when you are leading a community versus a single nation), as it confers a breadth of knowledge and ability to recognize the best in each community.

Is she perfect? No, but I would say that of all the characters, she is the most like Rand. The qualities you would call "Mary Sue"-like are remarkably similar to being "t'a'veren".