It looks like it can't be the thing we missed, because Rand already "destroyed [his people] with the leaf" by telling the Aiel that they used to be peaceful - that they, in fact split off from the Tuathaan and betrayed their vow of non-violence. Why? Because the Tuathaan are associated with the Way of the Leaf. Also, this moment caused destruction in that many Aiel ran away from their families and lives. It caused havoc by making them question what they've always believed about themselves.

It seems to me that this same moment may also serve for counting as the "slay his people with the sword of peace," because "peace" is also a motif of the Tuathaan. The "sword" here wmight symbolize the extra insult faced by the Aiel - they think they would never break an oath, but to find out they stopped being peaceful is as insulting as taking up a sword. Or it could be because Rand is a sword-user, and Rand told the Aiel of the "peaceful" past that they once adhered to.

I tend to write long-winded, so I shouldn't take issue with brevity, but would you like to express why you picked that passage? Or what you think it might reveal? Or how it could be an unnoticed thing that will play large in a later moment?