The thing about Azathoth, at least the way Lovecraft wrote it, is that its chaos isn't the kind you get in, say, a battlefield or a riot. It's chaos as the absence of any conceivable order. It's not a storm of randomness, it's the blank page before the storm was even a concept. The blind idiot god doesn't do chaos; it is the foundation that makes chaos possible, because it's the primal noise from which no pattern can ever fully emerge.
What gets me is how other mythologies often personify chaos—Tiamat, say, is a raging dragon, a force of violent disruption. Azathoth is none of that. It's a dreaming, mindless center, and the 'cosmos' is just the faint, meaningless music played around it. That's the true void: not empty space, but a state where ideas like 'thing' and 'happening' have no purchase. It embodies chaos by being the one entity that is utterly, perfectly indifferent to the very idea of meaning.
Honestly, sometimes I think later writers missed the point by trying to give it motives or a 'court'. The horror is in the lack of all that.