I agree with a lot of what you said, and found this to be an interesting read...

But it isn't like there aren't indicators for Egwene being LPD.

For one thing, I think you slightly miscast Latra's role in the Second Age. I don't think blanket opposition of the Dragon, or even a desire to control him, were really part of her M.O. She seems to have only come out in opposition to him when he proposed his Seals plan. But her role didn't end there either. For one thing, she seemed to be to female channelers what LTT was to the males, and for another, she completed LTT's job, once he was driven mad by the execution of his plan.

Some of these aspects rather closely parallel Egwene. Egwene's role as a uniter of female channelers, solidified in ToM, is what really gives me pause when I think about the whole rebirth issue.

If you look at Latra Pose, not as a controller or rival of LTT, but as the female representative of his "job", the check to the Dragon's power, the soul reborn to correct the Dragon's course, you can see how Egwene could fit this role. The implication would be that things went wrong in the AoL because these two didn't come to an understanding.

And we know the Pattern has other relationships where two Heroes are sometimes set in opposition and sometimes not. That's exactly what Artur Hawkwing told Rand about the relationship between their souls.

And this could also partly explain Rand's trouble with "women", that Birgette references. His lovers are one side of it, but cooperating and working with women also seems to be an issue for him.

Not that I'm sold on this idea though. I'd much prefer it if there aren't too many definite rebirths from the AoL around Rand. But whether there's any truth to Egwene being LPD, there's certainly some deeper connection between her and Rand that is yet to be revealed.