If you read my answer to Sidious which I wrote before reading yours, you'll see I noted many of the same points.

And I forgot one:

Since the beginning of EOTW a big theme for Egwene has been her thirst to learn. By the mid-series it has focussed on three angles: the One Power, TAR and politics. She has pretty much reached her own in politics now, she has surpassed Siuan because she is more daring and proactive and - I really don't want this to devolve into another debate on her achievements vs. her luck, she is now is a position to make good use of what she learned, far better than Siuan's position ever was. Her two main challenges in the short term are the male Aes Sedai question, and putting in practice what she's learned of being a political leader during a military campaign. She's worked closely with Bryne, very closely. That was her training to deal with the likes of Mat. She will have to learn to trust Mat and co. as much as she's learned to trust Bryne and not to overule him or interfere in his responsabilities but also don't hesitate to hold back the military when costs are too high, or their plan would cause impossible political problems. Bryne would have handed her the Tower sooner because there were better military ways to achieved it, but it would have been a Tower reunited in name only and impossible to rule and lead in the LB.

And there's TAR, but let's push that aside. Finally, there's the One Power. Egwene is just as motivated regarding her OP skill as Elayne or Nynaeve, and more "creative" when it comes to finding ways to use her knowledge practically. Cuendillar sold to solve the funding of the rebellion, Cuendillar used to block harbors etc. Egwene isn't the strongest of the "new Aes Sedai", but she's the most dextrous and the one who develop new skills the fastest. She's also cast the widest net, while Elayne focusses a lot of ter'angreal and Nynaeve a lot on healing. Egwene isn't one to pass on any useful source of knowledge, she rather solidly seize any of them within her reach. Where I'm going with that is that Egwene is bound to understand sooner than later what source of OP knowledge Rand truly is now. Not only that, but an early "impossible" goal of hers, which was for men and women to teach one another, is now within her reach. They need to link. We saw in the cleansing that Rand in a link could use saidar. He wasn't very skilled, but he didn't have all of LTT's knowledge and skills at the time either, in fact LTT vanished and didn't want to help with using the CK.... LTT might be clumsy when using saidar, far clumsier than an average woman, but no doubt there's still a huge lot he can teach a woman, guiding her with theory if not able to actually make a specific weave, and if anyone is good at puzzling things out from theorical clues, it's Egwene. She did it for cuendillar, she did it with Travelling - and that was from clues from a woman who was trying to hold back as many clues as she could, not from someone committed to tell everything he knows about how women in the AOL did this or that... Everything we know about Egwene points to the fact that as soon as she realizes mix gender linking is the key and that Rand has all the OP knowledge of LTT, she will bite and won't let go. Rand might think it's too late to teach the AS (I think personally that as soon as the Shadow's highly trained channelers outclassing AS and Asha'man alike enter the war, Rand will change his mind if he still stick to his mid series thinking that he has no time to teach the Asha'man himself), but it's inconceivable Egwene sees things that way. Egwene is one of the fastest learner, it makes sense she makes herself the student. The WT has the expertise to teach. Egwene has incidentally never linked with a man. I'll go on a limb and predict one of the momenteous "moments" left between Rand and Egwene will be when they link together. I will also point out that with the combination of her learning speed and her dexterity, Egwene herself is the most logical choice OP wise if Rand needs a woman for a specific task. Elayne/Aviendha wouldn't be bad either (Nyaneve is too specialized) but Egwene is better. On a side point, Rand will know best how to deal with the dreamspike, but there's no saying that just because he knows the theory he's skilled enough with TAR to accomplish what needs to be done. I think Rand will pass knowledge to the channelers to use in the LB (not that Egwene is so likely to want to learn just that, but they need to prioritize). His female student will be Egwene, who will teach enough teachers to spread the new knowledge fast to all those with the skills to learn them. The Green Ajah is a pool of massive potential. They are an army, they know strategy/tactics and discipline, but they're outclassed or at least very limited by their lack of advance OP knowledge. Egwene will remedy to that.

Rand knows Egwene well, and her thirst for knowledge. I even wonder if the necessity to teach the AS won't be part of the points he brings up at Merrilor, or rather if Rand won't press the point that instead of bickering the way to SG even though he will go on without them if he must, the WT could really use the little time left to increase their OP skills because Demandred's channelers will outclass them. I guess it's also a possibility the WT learn this the hard way, ie: see just how outclassed they are attempting to root out Taim with or without Logain's troops and failing miserably and be forced to retreat. They may not even have to attack the BT to learn this - Rand might also tell them with their current skills they don't stand a chance and if they try to attack the BT now (in the aftermath of the battle of Caemlyn, fo e.g.) they'll be wiped out because Demandred is involved and they don't have a clue in how much trouble they are if Demandred taught channelers to fight....

However it happens, Egwene would see immediately the need to learn - she has always jumped on such oppportunities, and her only hope to match what the Shadow has taught to Taim and co. is for the WT to establish a working relationship with Rand. Egwene might never see eye to eye with Rand on the breaking of the seals, but she can be very pragmatic at times. There's more than the seals in the balance, there's what comes after, when Rand does it even if no one understands/agree with him why it must be done. I'm not quite convinced that between the threat of the Seanchan and the threat of the BT Egwene will calculate the Tower is in any position to make Latra's choice. "Accept what you can't change, work on what you can change"... the WT needs Rand badly to learn to fight what the Shadow has put together. Logain's group is but a small taste of the kind of knowledge LTT can pass on.

Of course this is all also connected to the reunification of male and female channelers, another key factor where pragmatism and instinct of survival will have to triumph over 3000 years of prejudices, the sort of pragmatism and cool calculation of what was in the balance Egwene displayed when she took upon herself to organize Logain's escape.