Aiel do not face the draft. But an Aiel with the ability to channel would need to go to learn how to control Saidin. That Aiel would not be available to sing to the plants. I see no reason they couldn’t maintain the way of the leaf.
The Aiel really had two original seemingly unrelated tasks, follow the way of the leaf...
and care for the trees of life.* It isn’t clear to me when the Aiel started serving the Aes Sedai, one Aiel served Meirin before the Bore.
Originally the Aiel had few channelers, but in the last 3000 yrs, Randlandians culled the ability from their people and let countless women die prior to having children and hunted down all the men, most before they had more than a couple children. During that same period, the Aiel had women who lived hundreds of years mating with other Aiel, slowly increasing the channeler DNA in their population.
Upon DNA, the Traveling people (Tinkers) were started from the Aiel, they not only forsook their responsibility to the trees, they also forsook their promise to transport the magical objects. During the last 3000 years, they “stole” people’s children, aka brainwashing people into their crazy cult. Building dwellers didn’t like think because they would lose contact with their adult children in a world where people expected their children and grandchildren to live within walking distance and marry someone the parents knew well and trusted safe for their child. Thus the original redheadedness was eventually lost and the traveling people looked no different than Randlandians.
The People of the Dragon, I believe it may have merely been that once LTT became the Dragon and the Aiel served the Aes Sedai and LTT was the leader of all Aes Sedai...
Kinslayer, once the Bore was sealed, it would appear that Kinslayer merely retired (or continued to work, irrelevant) but he had kids and he got a huge palace, and his family lived with him. I recall somewhere it mentioning Kinslayer walking thru the hallways killing everyone in sight, movement, the weave doesn’t really matter. Ishy didn’t encourage Kinslayer, Ishy came after to gloat, torture Kinslayer by healing his mind so he could recall the murders. Then Kinslayer committed suicide and created the Dragonmount at that very spot. Rand was born near that spot.
While most people were not violent, some rare cases existed. One of the Forsaken tortured her patients. The Aes Sedai properly demanded she accepting the binding rod which is not very way of the leafy. It would seem the general population may be capable of a nominal level of aggression note the glimpse into the Age of Lengends when the man wearing a fancloak (old name for Warder cloak) knocked over the young Aiel Rand was riding in his memory. The man only apologized once he realized he knocked over an Aiel, behavior considered equivalent to knocking over a small child.
Fencing, while the rich could play this sport and we know boys generally wrestle, spar with quarterstaffs, these definitely seem too violent for Age of Legends Aiel and Tinkers today.