Like an orchestra, there seems to be several components to the Song needed to make the crops more fertile in the Age of Legends (as seen in the glass columns of Rhuidean, The Shadow Rising):

1. Ogier singers, probably Treesingers like Loial. Presumably the Tinkers have heard these songs and said 'That's not it." But it is a COMPONENT of the Song.

2. Aiel men singing, whom the Tinkers are descendents of, as are the present-day Aiel. The Clan Chiefs, including Rand know this COMPONENT of the Song, and presumably could sing it if asked to. If they could be brought to do so, after all, they have no desire to take part in Tinker affairs.

3. Nym directing the Song, like a maestro. The last remaining Nym, Someshta, is dead. Even if a new construct could be made, would it have knowledge of its part in the Song? Could Rand or another clan chief supply the missing knowledge?

Perrin has always been intrigued by the Way of the Leaf, and battled between what is necessary (violence, represented by the axe) and what should be (peace, represented by the hammer). He has finally chosen the Hammer for himself. Now he must try bring an end to the larger scale violence or his personal choice will be for nothing. Around him, Aram took up the sword when Perrin first took up the Axe. Can he turn Aram back to the Way of the Leaf, the Way of the Hammer? How could he do this?

He could bring the Aiel to the Way of the Hammer. The Aiel believe there will be naught left to them after the Last Battle but ‘a remnant of a remnant,' that they will die in battle serving the Car'a'carn. But Perrin could find a better use for them. If he could convince them to sing the song they learned in the glass columns, they would have a new purpose: Making green things grow. Bringing bountiful food to the world. Though the Aiel would not turn to the Way of the Leaf, like the most-hated Lost Ones, they could go halfway, and follow Perrin's Way of the Hammer, building the world up from the wreckage of the Last Battle. If an Aiel still follows ji'e'toh, I imagine he would find much ji in making things grow for others, especially having lived in the Three-Fold Land where there was so little. What need for dancing the spears if there is water for all?

The ‘remnant of a remnant' would mean the Aiel are not necessarily destroyed in the Last Battle, but only the memory of that other memory from the Age of Legends would define their future existence.

Why Perrin and not Rand? Obviously Perrin is the only one with much contact with Tinkers, and he is dealing with the Aiel, as both allies and opponents. In fact, Aram might be the key to the whole process, as convincing him to choose the Hammer will be more difficult than getting the Aiel to give up the spear in favour of singing.

This theory covers some aspects of Aiel prophecy, Min's vision of Green plants flowering around Perrin, Aram's purpose, the riddle of Hammer and Axe, and the secret of the Tinker Song. I know there are some loose ends, but I'm sure you can find some I haven't considered, or patch some of the holes, like where to find a Nym.