@Ant
Yes. The issue is with the fact that Azathoth wouldn't do anything, nor would anything actually happen to the Outer Gods themselves. "Never" isn't really the proper term for something beyond all forms of time, but they simply never would have existed.
It'll probably be easier to explain why this is such a pain to properly quantify if I post some quotes.
Time, change, and motion on all levels are illusion.
"Time, the waves went on, is motionless, and without beginning or end. That it has motion, and is the cause of change, is an illusion. Indeed, it is itself really an illusion, for except to the narrow sight of beings in limited dimensions there are no such things as past, present, and future. Men think of time only because of what they call change, yet that too is illusion. All that was, and is, and is to be, exists simultaneously."
The Outer Gods do not experience this. They view reality as totally unchanging, because that is what it truly is.
"After an impressive pause the waves continued, saying that what the denizens of few-dimensioned zones call change is merely a function of their consciousness, which views the external world from various cosmic angles. As the shapes produced by the cutting of a cone seem to vary with the angles of cutting—being circle, ellipse, parabola, or hyperbola according to that angle, yet without any change in the cone itself—so do the local aspects of an unchanged and endless reality seem to change with the cosmic angle of regarding. To this variety of angles of consciousness the feeble beings of the inner worlds are slaves, since with rare exceptions they cannot learn to control them. Only a few students of forbidden things have gained inklings of this control, and have thereby conquered time and change. But the entities outside the Gates command all angles, and view the myriad parts of the cosmos in terms of fragmentary, change-involving perspective, or of the changeless totality beyond perspective, in accordance with their will."
Keep in mind this is not merely referring to time and change as we know it, nor to purely physical dimensions. It also includes their metaphysical analogues, because those beings as well experience their own forms of change. As an example, the realm of 'Umr at-Tawil is mentioned by name, and referred to as a "small wholeness" and "infinitessimal thing".
"The world of men and of the gods of men is merely an infinitesimal phase of an infinitesimal thing—the three-dimensional phase of that small wholeness reached by the First Gate, where 'Umr at-Tawil dictates dreams to the Ancient Ones. Though men hail it as reality and brand thoughts of its many-dimensioned original as unreality, it is in truth the very opposite. That which we call substance and reality is shadow and illusion, and that which we call shadow and illusion is substance and reality."
As I said before, this limit of "change" applies to metaphysical/transdimensional analogues of normal dimensions as well, for said "infinitessimal thing" was already beyond the infinite spatial dimensions of physical reality.
"Memory and imagination shaped dim half-pictures with uncertain outlines amidst the seething chaos, but Carter knew that they were of memory and imagination only. Yet he felt that it was not chance which built these things in his consciousness, but rather some vast reality, ineffable and undimensioned, which surrounded him and strove to translate itself into the only symbols he was capable of grasping. For no mind of earth may grasp the extensions of shape which interweave in the oblique gulfs outside time and the dimensions we know."
This is the issue with the Outer Gods. They do not experience any form of change. They do not fight. They do not have a limited sense of self. They simply are, like an unchanging idea. They would not be without Azathoth, just as everything else wouldn't. However, while the family tree we are given is true, it does not mean the same thing as a traditional family tree. Something like "Azathoth begat the Mist, who begat, Yog-Sothoth, etc.". It's more akin to "These things have always been, and they are because Azathoth".
tl;dr It's ******* complicated.