Atheism and Nihilism

Atheism does not imply nihilism. Nihilism does not imply atheism. The two are distinct, but there is a popular perception that the two are tethered together. Why does this perception arise?

The first reason would be the historical one, which is that in Christendom morality was rooted in theism, so a rejection of God implies a rejection of morality. This is less interesting, since I wonder why the perception arises still today. You can say that it is a holdover from the past, or you can say that the perception only arises among those who still root their morality in their theism, but I think there is an epistemological reason.

We do not need a God for true, objective morality to exist. Perhaps it is just there. Perhaps moral imperatives are just natural facts. Yet, if we reject God on the basis that it does not satisfy the intellect to believe on such scant evidence, would we not also have to reject any kind of external morality for the same reason? It is the standpoint of doubt that causes trepidation: if your doubt causes you to reject this picture of reality, what else is missing from your picture?