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.

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".