I can feel your anger

[Skepsis]

I don't find this very convincing. People don't look to theism for a cornerstone either. Nobody's actions are informed by the mere proposition, "God exists".

Don't really see why you don't understand. Theism, the positive belief that God exists, informs the belief sets underneath that positive claim directly. They give meaning to an otherwise meaningless set of beliefs.
For example, "God created the universe", a Christian claim, wouldn't mean anything unless you believed in God.
I feel like I am explaining to you the definition of "of". It's hard as hell to do, but represents the most basic of ideas.

My point is that people are informed by sets of beliefs, and that when those sets are theistic, we say that theism informed their actions; why isn't that when those sets are atheistic, we say that atheism informed their actions?

Theistic sets make a positive claim that other ideas can use to be given meaning. Atheism is a lack of belief that cannot and does not mean anything to the positive beliefs that simply don't incorporate God.
You would have a single category for theism, and infinite categories for a-unicornism, a-Cthuluism, atheism, a-conspiracytheoryism- you get the point, I hope.
Atheism influences the beliefs of a Nihilist just as a-unicornism does, y'know?

Here's the big question:

What's wrong with saying that if a person's set of motivating beliefs lacks the belief, "At least one god exists", then their motivating set is atheistic?

Like I explained, it is silly to try to say that lack of belief is the cornerstone of affirmative belief.