Is nihilism the logical extreme of atheism?

[Thumpalumpacus]

As I see it, atheism undermines all three. And without that solid foundation, all atheists are tacit nihilists no matter how adamantly they deny it.

Perhaps that was your atheism. It certainly isn't mine.

One of the actual joys of atheism is defining your own purpose in life. Such joy is an emotional response that doesn’t rationally counter nihilism. When someone defines purpose as that outcome towards which something is directed, then they are invoking Final Cause. Atheism, per se, does not exclude final causes, but the reduction of the world to purely efficient causes acting on material bodies does. Therefore ‘purposes’ are illusions born of viewing higher-order processes that are fully determined at lower levels of order. So while it would appear as-if intelligent agents have goals, in actuality there are no final ends and it is irrational to speak about any life having purpose.

The unquestioned premise here is that all atheists are materialist reductionists.

When people talk about a life’s purpose they usually are thinking of a higher criteria that just goal-seeking behavior and final ends. What they really mean is that their life counted from something, i.e. their life has, or will have, lasting value. Value requires that someone appreciates and desires something which is valued. The value of life for the person living it, seems self-evident, since all other valuables require already having a life. But because life ends, the lasting value of a person’s life depends on their life having continuing value to those remain alive and future generations. Then the sun blows up and with it any value our lives once had. Thus at a bare minimum, for human life to have value there must be some enduring agent to whom human life is valuable.

Death doesn't render life meaningless, that is a silly claim. You may as well argue that the period at the end of this sentence removes meaning from each and every word in it.

Your unquestioned premise in this segment of your post is that for meaning to hold true, it must be eternal. When you consider that the brain which ponders meanings is itself not eternal, you'll understand your own absurdity here.