The big problem with Egwene is all major characters except her got schooled about not being as great as they thought they were

I believe she may be Eldrene reborn (or a descendant), and they might even have used the same weave.

 

I also believe that the Emonds Fielders are all bound to the Wheel like Rand and the Heroes of the Horn are. I think Mat is usually a commander, or even the Hornsounder when it's blown. Not every Hero can be in the Horn. Someone has to blow it. Someone has to fight those battles before it's blown...etc.

There's one major problem with Egwene being Eldrene reborn, or Mat (or anyone else from Emond's Field) being reincarnations of people from Manetheren.

 

The people who are reborn again and again throughout the turning of the Wheel, living new lives and becoming new people, are very specifically the heroes who have earned that privilege and been permanently bound to the Wheel itself: Birgitte and Gaidal, Rand, Artur Hawkwing, etc. Meaning if Egwene was anyone reborn, she would have been called back as a Hero of the Horn when Olver blew it, like Birgitte was. There's no sign or hint of her presence when it's blown.

No. Anyone can be reborn, but people bound to the Wheel will always be reborn whenever their particular skill set is needed.