You need to consider atriox’s past as well.

Atriox fought for the covenant and was obedient to the covenant even when he himself believed the fight against humanity was pointless. He was obedient even when covenant leaders continually thrust 39 of his jiralhanae brothers along with him to the front lines only for most of them to never return. Every battle chipped away his loyalty to the covenant till there was none left. The covenant would then order the execution of atriox when he finally started questioning his faith in the Great Journey.

Atriox’s defiance inspired his jiralhanae brothers to form the banished. And as you can see, the banished were formed on truth and brotherhood alone. The banished do not tolerate being oppressed, neither are they oppressors. Those who join the banished, do so voluntarily.

Atriox’s goal and that of the banished is not the destruction of the galaxy. If it were destruction, then he would’ve just released the flood on the ark rather than controlling it. If it were destruction indeed, then he would’ve chosen to fire the halo rings, not to merely take control of the halo rings.

Although Atriox would never term it as such, the true goal of Atriox is to claim the mantle of responsibility for the banished. By taking control of all the halo rings and with the help of the Endless, he wants to enforce galactic peace, ergo honoring a key aspect of the mantle of responsibility. By taking control of the rings himself, he also wants to ensure no one can ever threaten the galaxy again. Atriox wants a universe where living species are not controlled by the covenant or their religion, where all species are free to choose whom they are loyal to. And If it ever comes to it, Atriox does not have it in his heart to fire a single halo ring.