What that whole future thing has always meant to me is one of those dystopian possibilities that appear so frequently in sci-fi & fantasy TV shows (due to its length, WoT shares some of the properties of such works), where the characters or audience are given a glimpse of the dire fate in store for the world if the heroes fail. That's all this was - something to raise the stakes and make the readers care about stuff beyond Rand's fight with the Shadow (against which almost all other issues justifiably fade into insignificance). Even so, Aviendha's forewarnings hardly justify taking any measures that might risk the outcome of Tarmon Gaidon. No matter how sad the fate of their umpteenth grand-daughter, it's still a better world than one where the Shadow wins.

That being said, regarding the whys and hows of the situation, particularly the absence of the mothers, there are a number of educated guesses one could make. One that I keep coming back to (though I haven't the faintest idea how RJ & B-Sand would get there) is RJ's assertion in his blog that Excaliber was his favorite genre movie, particularly the final scene. Said scene features King Arthur's body borne away on a boat accompanied by three women - an image foreshadowed more than once in WoT. So it could be as simple as Aviendha, Elayne & Min going off with him in RJ's homage to that film. Other explanations could be his death or need to otherwise leave the world, which is why his daughter doesn't seem to remember him. With Rand gone, that would provide a fair excuse for the apparent negligence of his children's mothers in raising them (Aiel customs aside, I cannot see Elayne & Min overlooking the fact that their children are not just sons and daughters of friendly mothers, but are actually brothers and sisters with one another - yet there is no sign that Aviendha's children consider Elayne's or Min's to be their siblings, and for Oncala & Talana, the relationship of their grandmothers seems more important than actually sharing a grandfather). Plainly, a large part of this grim future is explained by the estrangement of the Aiel from their erstwhile wetlander allies, and the estrangement of Aviendha's children from their wetlander half-siblings no doubt contributes to their roles in causing the Aiel to break ranks and incite the fatal conflict, and that estrangment is hard to envision given the portrayal of their mothers' relationships and the amount of attention and effort the Wise Ones put into forging ties with Elayne and, to a lesser extent, Min. Obviously the moms have screwed up and that's not so hard to understand either. Grieving widow(er)s in WoT have a long history of horrible decision-making:
- Lews Therin incinerates himself and not only raises a volcano but creates an island that will no doubt contribute to the Aes Sedai myth. He also blows off all the good he could have done as a now-sane male Aes Sedai of authority in the world by killing himself over his grief.
- Eldrene destroys the most beautiful human city in the world in her grief over Aemon's fall. Ironically, he was able to hold out so long and kept fighting a doomed battle in the hopes of preserving a nice home for his people to come back and live in. Thanks to his widow's thoughtlessness, the blood of the men of the Mountain Home would forever dwell in the flat woodlands and swamps that were left after her rampage.
- Artur Hawkwing, probably the greatest man of the Third Age, went on something of a tyrannical bender until he was resupplied with nookie.
- Morgase Trakand indulged in a relationship that was, judging from its outcome, highly unhealthy and unstable. It ended with her sending an army to hunt him and him handling her in a way that would guarantee her being granted a temporary restraining order on short notice in modern society, and a decade and a half later, people are still afraid to mention her rebound fling's name to Morgase.
- other such evidence of loss of a sexual partner inducing mental derangement includes volunteering to go along on a Forsaken-hunting trip you wanted no part of the day before, quitting your job as clan chief and letting a definitively evil guy kill you because of a passing resemblance to your late boot-knocker, assassinating a king and plunging a country into civil war before courting execution or torture by getting drunk on your archenemy's stomping grounds, sleeping with your late husband's successors before inducing them to lead a genocidal rampage into the wetlands, hopping on a boat to Salidar to become one of the most troublesome novices in their history, and handing off your late husband's epic fantasy series to thirty year old Mormon with a cripplingly limited vocabulary, poor reading-comprehension skills and maybe three published grown-up novels under his belt because he wrote a nice tribute on his blog.

Given the track records of WoT people mouring their love interests, we should be glad Aviendha's kids were allowed to grow up and never mind how badly they might have been raised.