The people known as the Tinkers were the first group to split off from the Aiel after the breaking. They held onto some of the basics of their culture, the way of the leaf, while discarding their Age of Legends tradition of service to the Aes Sedai. They broke off after a couple generations spent hauling the items of power that were bestowed on them by the Aes Sedai. That was their final assigned task during the breaking.

The Jenn never "failed" the Aes Sedai. They were loyal to their charge to the very end. They chose to enter the waste for whatever reason (most likely safety from humanity as most people would seek to avoid such harsh conditions). The rest of the body of the Aiel outcasts followed the Jenn's Wagons out of a combination of tradition(they had been doing so for generations alread) and dedication to a twisted memory of what they had been.

The twisting of stories over time is a common theme in the Wheel of Time and we often see their effects.