I don't equate atheism with nihilism, although I am convinced that a logically consistent atheist will become a nihilist. In answer to your question, God is self-existent. That's part of how theism defines the term. Inherent in that is the assertion that purpose is not a term that can be applied to God. Purpose (objective) is always assigned. At least I cannot think of a single counter example to this. Perhaps you can, and I'd be interested to see your reasoning if you do come up with one. So the question becomes 'who could possibly assign objective purpose to the Creator's existence?'. As I'm sure you know, trying to do that leads to paradox. Since God is the ground of all existence, there can logically be no one to assign purpose to Him.
Humans, on the other hand are not self-existent. We are contingent beings, owing our existence to something beyond ourselves. God has assigned a purpose to us-- we are to bear His image. That is the reason for our existence. The logic error non-nihilistic atheists make is pretending this doesn't obtain in their case. Ignoring the fact that a contingent being cannot assign objective purpose to it's own existence, you guys try to lift yourselves by your own bootstraps. At this point you become illogical. You actually employ a nihilistic assertion of naked will to create meaning for your existence apart from any rational considerations at all. You are, of course, free to do so. The trouble is, and history bears this out, your private purpose does nothing to ensure that others should recognize that purpose. When confronted with injustice wedded to power as in Stalinism the atheist has no moral authority to challenge the right of the State to do whatever it deems fit to ensure its own survival.
Private purpose (and morality) can work out in a culture that still operates on a predominately theistic paradigm, but when that disappears you are truly left in the jungle. Humankind then separates into two groups: sheep and wolves. Sheep make rules to protect themselves from the wolves, but the wolves certainly do not feel bound by such arbitrary precepts. Power becomes the ultimate coin of the realm. I'm sure you're perceptive enough to see exactly that happening in the Western democracies. We no longer try to persuade one another regarding social policy. We battle through the courts and legislative branches of our governments. Power, not persuasion, is how we now make social decisions. I am convinced that the reason for this is that we no longer share a common understanding of our purpose. Your world of private purpose is bearing poisonous fruit.