The soul split in three is a stretch. We know RJ conceived of souls as immortal entities, reborn over and over but not stagnating in their progress, not devoid of free will during incarnations etc. People conceive of the Wheel as this giant puppet master engineering everything down to the minute details largely because three main characters are ta'veren and more constrained than others (and the Wheel using them to have greater control of certain aspects of the lives of people around them in order to weave the pattern toward some specific situations), and because Rand is also a Hero of the Horn, reborn to accomplish a specific mission, which happens to be similar to the mission he had to accomplish in his previous life.

But Rand is a wholly different man than LTT was. His biography is different. His personality is different, and it's only by the mid-series after already having travelled a rough path and when his past life started to leak into his mind that Rand became more similar to LTT (and as it turns out, a major theme of the latter series was for Rand to remember himself, to bring peace to LTT and move on). It's the same from Birgitte, and in fact much of her experience seems to be there for contrast, to show us that what was happening with Rand was not normal. Birgitte has a few similar traits in each incarnation, dominant traits to her soul that made her chosen by the Wheel to become one of the Heroes, most likely. Her "eternal" ties to Gaidal aren't the norm, they are the exceptions, the result of the Wheel choosing this dynamic to be one of its corrective tools, and blocking it in place by making both Gaidal and Birgitte heroes. Beyond that, she isn't always that blond archer.... We know for fact her incarnations were massively different, not near clones of one another, going through very similar events, replaying the same patterns with the same people (except one way or another she's pair up with "Gaidal", but that's because they're both Heroes). No, Birgitte was once a lowly soldier in the AOL, played a role in the founding of the Tower, was one of adventurers attracted to the court of Hawkwing (that's the incarnation where she was this blond archer named Birgitte, and the source of the Birgitte stories from the Great Hunt cycle) and is now Warder to one of Rand's lovers. Wholly different lives, wholly different crowds, and even in regard to Gaidal, different stories.

The existence of ta'veren and Heroes of the Horn proves that the Wheel needs special tools/devices in order to gain a real direct control over people's lives. And even then... it doesn't control everything... the Wheel isn't even sentient, it's a purely logical system like a computer. It takes everything that is, say, Rand, into account, and influence him through an input/feedback process, gradually bringing him and people around him to certain situations that must happen. If the Wheel could so easily bring people together, take over their free will completely, it would not need ta'veren or Heroes... The process seems rather that the Wheel "stores" data (that may be what the Pattern of Ages is) about every aspects of humans and experiences, so it can calculate probabilities that this or that happens. In this situation, someone like Rand might choose like this or that, and as "this" is needed, placing that other event on his path increases the odds that he will choose "this". If it fails, the Wheel re calculates and outputs something to bring Rand back to "this". That's how women see things in Rhuidean: always tons of options and possibilities, occasionally the choices are very constrained or the consequences of not choosing one over another are dire for the person. Even Mat had choices at some point. It was either continue on the path and in this case it brought him to Rhuidean, while every other choice that would make him avoid that lead to his death. Mat seems to be the most difficult to control, and Perrin the easiest, with Rand growing more and more difficult through the series. With Mat, it seems the Wheel is constantly having to recalculate its projections and put new stuff in front of him to try to bring him back on track. I think this may be where the "dices" come from. Mat is stuck with them rattling in his head until he takes the right path. Annoying, but effective. It may have to do with SL, as Padan Fain who combines both SL and TP corruption has, according to Jordan, to escape most influence from the Wheel.

So, it sounds unlikely in the extreme that Rand is surrounded by a cast of people who are the rebirths of specific people from LTT's entourage, replaying together a variation of the same theme... The only possibility that such a thing would be likely is that there are some hidden Heroes among the characters, with relationships to the Dragon soul similar to the Gaidal/Birgitte one - but there's been no hint of that so far, and in fact Birgitte jested about the Dragon soul and his problems with women, which suggests he's not tied to a particular one, but how much he falls in love and how it complicates his life as Hero is one of those "dominant traits" to his soul that manages to come out through his rebirths. He seems to have three this time around because the Wheel has calculated that he would again be destroyed if by one's death and the other two are there to save him and make him hang to life if that happens. The "extra twos" are there to correct a weakness in the Dragon soul, that makes him extremely affected by loss, especially of women, but as we know from his epiphany, the Dragon's soul ability to love as he does is also precisely one of his main strength, absolutely required for him to be able to be the Dragon. It's thus a matter of avoiding the bad sides (like how much the soul is affected by loss) so the good side (the immense ability to love, and to project his into a very deep love of life and belief that humanity and the Pattern must keep going) of this aspect of Rand can play out.

If the Wheel could predict the probability that LTT's life would leak into Rand's (and in theory it had the data for this, from male channelers born since the Taint), it made the "extras" even more logical as the leakage meant Rand could become very affected by Ilyena's murder, through LTT - and two backup wives he also loved equally wouldn't be a luxury should anything happened to one of them, and to help Rand live through the loss of other women as well. Another thing the trio does is that it made it much more difficult for Rand to put his wife aside, sheltered somewhere. It offered the Wheel many more possibilities to keep "love" present in Rand's life at every juncture. He'd push Elayne away, enters Aviendha. He manages to leave Aviendha behind, comes Min. And so on. Rand has tried to leave love behind him, out of love and out of fear of loss, and thinking of this as a weakness the Shadow could exploit, but as we know since the epiphany, the Wheel had no use for a Dragon who has managed to put love behind... as a Dragon without love is a Dragon who would loses touch with humanity, and the Shadow helping a bit that results in a Dragon who develops the same vision of the Pattern and Life as a prison for souls, an hell of eternal suffering through rebirths as... Moridin.

I don't think it so likely Elayne is the rebirth of Ilyena, in fact it would seem counterproductive and plain idiotic that the Wheel would decide to put near him the woman who would increase the danger that Rand fell into obsessive guilt over the Kinslaying. If Jordan intended to go there, LTT would have already recognized her most likely, not with a final answer probably (Jordan never wished to give us final answers on "cosmic" issues like this, it's not knowledge the characters would ever possess, and it's not from this "omni", all knowing perspective he wished us to experience the series). All that we rather saw happen is stuff like Elayne's hair causing grief to LTT as it's the same as Ilyena and brought back painful memories, and these emotions affecting Rand negatively.

It would be more likely Egwene could be Latra Posae reborn if the majority of Aes Sedai didn't act either like they were, or attempting to find ways to surmount their fears and take the opposite path to LPD! They are all the heirs of the Fateful Concord.. Green Ajah, Red Ajah, Amyrlins... The Greens for e.g. would like to stand as the opposite to the Concord, but deep down they all have to fight atavic fears to let the Dragon make the calls. Even Cadsuane burned her fingers with that, for all she tried not to interfere with Rand's plans and only work on his personality. The Reds all seem to see the Concord very positively, and Elaida in particular acted as if this time around she intended to triumph at imposing her way to fight the LB where LPD has failed. Egwene is torn between her desire to support the Dragon (and her former friend), her own fears of what will happen if she does and all the advice she gets not to let the Dragon make the decisions. Furthermore, all Egwene has heard of for many months are reports about a man who was no longer acting or making decisions like Rand would, and by all accounts had become a dark, temperemental and often illogical man.

In fact, if anyone is Latra Posae reborn in the series (but I think it's unecessary - the whole idea is rather than the Aes Sedai are still very affected in their worldview by what happened within the organization during the WOS, and by the Breaking, and inconsciously divided between pro-LPD and anti-LPD and every shades in between the two extremes), it would most likely be either Siuan or Elaida, and their grand plans to ensure the LB would be fought as they thought, not as the Dragon would, are them. Siuan thought to make Rand dance to her tune, and when she failed, she became the greatest influence near Egwene encouraging her to be extremely careful about Rand, fuelling her suspicions, and inconsciously trying to make Egwene follow her own vision and old plans. And both have been pushed aside, both by Egwene. Now it's up to her to keep an open mind and finally surmount her underlying rivalry with Rand that has long predated their current positions/jobs, and up to Rand to come up with more convincing arguments to get the female Aes Sedai to support his plans that LTT managed to do (because one side of the causes of the Concord in the first place is that LTT's plans was fundamentally flawed and too dangerous. So was LPD's alternative down the line. The AOL Aes Sedai just didn't manage to solve the problem of sealing the Bore. One plan lead to the Taint, the other plan involved using doom day devices and so much power that if the DO managed to break the shield created it likely would have destroyed the Pattern. It sounds like the AS could not think outside the box. The drilling itself had shown that OP/TP interactions were immensely dangerous, but neither LTT nor LPD could think of any plan that would take this into account. Rand is aware of this now, aware his failure came from touching Shai'tan with the OP. We'll see what he comes up with. Right now it's perfectly reasonable for Egwene to have reacted so negatively to Rand's announcement, all he gave her was "now I will break the Seals and this will make the DO all the more powerful, but we have no choice to do this as if we delay any longer humanity will no longer be able to fight when the inevitable happens", but nothing at all about what he intends to do after. He won't begin to sway the White Tower or Egwene before he present them his plans for the next steps in the LB after he breaks the seals and starts it, or provide them with extremely good reasons why they should trust him and take a leap of faith with him. That will start at Merrilor (and conclude or not, we'll see), for now Rand's intent was to shake the Tower big time, and forewarn the Aes Sedai the time for the important decisions and discussions of plans had come, and then let them do the hard work of bringing as much of the world's leaders to this meeting as possible - to support his plans (which most are not competent to judge) or to try to convince him to change his mind matters not for now, it's at Merrilor itself he intends to make his case. Not sure what he intends to do in the meantime (beside designing his plans for SG...), though I suspect "new Rand" may not have the same view of the Seanchan that "dark Rand" did and he may intend a new approach to convince Tuon to also come to Merrilor, similar to the one he took with the White Tower. After all, that's the resolution of the Finn's riddle: Merrilor brings the north (Borderland rulers) and east (the White Tower group, including all the eastern leaders gathered behind Egwene.. Elayne, Darlin and so on) together. The Seanchan has brought together the South and East now. Now Rand must find a way to bring the two camps together "as one" at the Fields of Merrilor.