I have often commented with what looks like to Jews as an "obsession" with what one "believes" (or doesn't believe, for that matter). Your comments are the closest I've seen to what I've been trying explain.
Belief is merely a static, mental assent to a concept, usually an abstract one. Belief in God is such a thing, and even disbelief in God. It's all about whether or not one grasps an abstract concept of something transcendent in the mind or not.
Gentiles will argue, debate, and literally get violent over what one claims they do or do not mentally assent to. To many Jews this is ridiculous.* It's like how denominations draw membership lines based on what you "believe:" How can you ensure that a member properly grasps an abstract theological definition of God (take the Trinity as an example) in just the right way as to warrant membership and salvation? And isn't the theological definition always claimed to be limited by human frailty? How does demand that one assent to an abstract and inefficient theological concept as a requisite for communion and salvation if there is no way to validate that the claimant correctly mentally grasps the incomplete yet transcendent abstract concept?
And then, what big thing is it to say you don't believe in such a concept? If you can't prove what one truly mentally assents to, then how can you prove what they don't? And even if we could, what great accomplishment of proof of logical superiority is not exercising the mind to grasp an abstract belief? One might argue that not attempting to believe suggests mental laziness more than mental efficiency.
No, I'm NOT saying that atheism and theism are stupid. What I am saying is that what you mentally assent to and what you don't matters little if it doesn't affect the world around you in a positive way. One can claim belief in the most virtuous of gods and doctrines and still not be virtuous themselves. Likewise someone can be free of religion and be greatly just, honest, and not merely sanctimonious about it either.
Even in the Gospels, Jesus commends righteous living over true doctrine. The Good Samaritan proves to be right with God even though he, as a Samaritan, holds doctrinal beliefs that are wrong compared to those of the priest and Levite who pass by. (Luke 10:25-37) And the Eschatological Discourse of Matthew 25:31-46 has Jesus granting eternal life to those who serve strangers, the poor, criminals in jail, and other marginalized members of society as they would Jesus, not to those who make more converts to Christianity.
The point is that somewhere we may have become lost in the "belief" game, debating and attempting to convert others to our "way of thinking," while never being intelligent enough to consider how impossible it is to ensure how another does or doesn't grasp abstract concepts like we do. It's a waste of time, even for "believers" as Jesus illustrates.
Your illustration of dealing with children is perfect and can be applied to all. It's not what you claim you believe that matters as much as living out the best of what you believe in your treatment of others.
*Footnote: To be fair, some Jews have been as obsessed with the "belief" argument to the point of violence as well. It may be far less apparent however due to belief not being a requisite for status in Judaism, but Jews cannot claim complete freedom in their history from the problems caused by fighting over the issue of mental assent.