The Aiel don't interbreed... The Tinkers have the Way of the Leaf, the Aiel have the pure blood of the Da'shain. Between the two stand Rand and Aviendha, and quite possibly Perrin who has Gaul (a future clan chief if the WO have their way) and ties to Rahen and the Tinkers. That he/Gaul/Rahen contribute to the reunion of the Aiel and Tinkers could very well be what the viewing of Perrin standing among flowering trees is all about (flowering is the harbinger of the Trees bearing fruits...).

Aviendha has seen what the future holds if the Aiel don't abandon the warrior way post TG. She's not bloody likely to do nothing about that! Rand has seen the gathering of the Tuatha'an in Ebou Dar, the Tuatha'an have even returned to a life where rather than standing aside uselessly they live among the people and provide service (mundane and perhaps eventually spiritual) and now he has LTT's knowledge Rand knows all there is to know about the Da'shain Aiel, and about Singing. Apparently, LTT or his wife prized Singing a lot, and everyone with the Talent was welcome in his house... And there's also Elayne's AOL library, and her "i-pod". Plenty of possibilities, and the "devices" in place to make it happen fast. The AS of the post-SG AOL (early Breaking) tought it vital that the Da'shain survived and they sent them away. The Rhuidean AS seemed to think vital that the Aiel survided, but also that the Way of the Leaf survived.. they just didn't understand they had to become warriors to survive until then. Back in TSR/TFOH, Rand took the first step by revealing to the Aiel what "their sin" was exactly.

At the beginning of TOM, Rand suddenly turned into a kind of human Nym, plants suddenly thriving where he walked... the fourth Age verses we have present the Dragon has someone who made things grow. And note this all began in Ebou Dar, where Rand "lost and blind", at the bottom of the barrel took refuge among the Tuatha'an who appeased him briefly. It's this experience that led him that made him realize in some ways the Seanchan have had more success than him at creating order and safety, that the Tuatha'an had sought their protection, not his! It shamed, angered and despaired him - he almost destroyed the whole Seanchan leadership with balefire in spite.. which drove him to Dragonmount and his epiphany...

It's all coming together... you have to be blind not to figure out Singing one way or another with be a feature of the series's ending.

As for Singing during the LB itself, as I said I'm totally undecided if this is something Jordan put in for TG or only for after. The "Guide" long ago alluded to the fact there may be more to the Singing Talent than the Crop Songs. We know the Da'shain songs appeased the minds of a mad channeler for a while, it suggests it has a powerful effect on people. It could go either way.

The fact the Tuatha'an have twisted everything and turned the old songs of the Da'shain into a single mythical Song that would miraculously bring a new Golden Age is irrelevant. Rand knows better, any Wise One or Clan Chief know better. They all knew what the Da'shain really were, Rand most of all - and we still don't know the details why the Da'shain were called The Children of the Dragon. It's all a matter of the Tinkers and Aiel coming back together (which has started subtly happening long ago in the background.. hundreds of Aiel asked the Tuatha'an who had taken refuge in the Waste to be taken in and learn the Way of the Leaf back in TFOH...), which we can surmise from Aviendha's visions is interconnected with bringing the Seanchan, now the Tinkers' protectors (and ironically, the Aiel's most bitter enemy), to the Light's "alliance".