well. i'd dispute that claim. way too many of us have only the slightest clue about how to reason effectively. just because someone has rejected theism doesn't mean we can generalize their ability to reason effectively across all other areas. take a stroll around the forum and see how few people are able to accurately spot logical fallacies. ask most atheists what "confirmation bias" or "motivated inference" means and they'll have no idea. what are Bayes' Theorem and inference to the best explanation and how should we apply them in the service of skepticism? yeah...
and i'm not saying i'm some pro at it or anything. but i know enough to know that most don't.
but, anyway, to your question, it's a useful tool because the same rules apply to everyone. so when you say something and allege it to be true logically anyone can apply the same rules and see if it measures up. that means that when we say that something is logically true we mean that it is universally true for everyone, everywhere. powerful stuff.
these things allow us to account for our cognitive biases that otherwise rule our reasoning and decision making.
What's the alternative? You mentioned three things and them refer to them as "it" in the singular. Reasoning is one thing. It uses empirical objective evidence and logic to reach conclusions. The beauty of logic is that given true premises and a valid argument, the conclusion is deterministically true.
We don't always get evidence that lends itself to deductive logic, in which case we resort to inductive logic. We use induction a lot more than deduction. In induction we look at a set of data and impose an explanation that makes sense of it.
The short answer to your question should be obvious: it works.