> Another frustrating aspect of human nature is that most people don't know how to be logical.
Many human philosophies reject logic (Or any mechanical system of derivation) as a valid epistemological device. Logic in the west is heavily shaped by Greek philosophers, like Plato and Aristotle, and then Latin Catholicism's embrace of them in the medieval times (logic is not particularly important in the eastern churches). Without the effort of the Church in the West, logic would not exist as a common form of 'proof' today. In Arabia, Greek logic was embraced by the Muslims. But, it is not a universally embraced philosophical framework.
In fact, there's not even any such thing as 'logic'. There are multiple logics in Western thought (and in mathematics and philosophy and even computer science) that are often at odds with one another and cannot be reconciled. Western philosophers like Godel even admit that logic cannot explain everything.
Of course 'logic' (and it's now in quotes to refer to your conception of it, which we've already established is ill-defined given the number of extant logical systems) in normal discourse now includes the law of excluded middle, which undermines the entire system, and is incompatible with logic's own embrace of rationality.
Let's take a quick tour around the world though.
Several branches of western philosophy do not admit logic (the way you think of it at least) as a valid epistemology:
1. Mathematical Constructivists reject universal applicability of the law of excluded middle. In the intuitionistic logic they use, the phrase 'P is either true or not true' is not one that can be assigned a truth value. This is not to say there are no propositions which one could argue successfully are individually true or untrue, they just reject the universal applicability of the law. This is really important, because in 'formal logic' the way you conceive it, the law of excluded middle makes the entire logic unsound.
2. Pragmatists (John Dewey, Charles Sanders Pierce) often reject logic outright (https://en.wikipedia.org/wiki/Pragmatism#Logic).
Many branches of non-western philosophy do not admit logical inference as fully valid, or augment it such that it no longer fits with western conceptions.
1. For example, the Charvaka philosophy from India rejects logically inferred knowledge as always doubtful. In that philosophy, the only unqualifiably true knowledge is that which one observes. Studying 'logic' in an attempt to be 'rational' is not useful in that system: https://en.wikipedia.org/wiki/Charvaka.
2. The Vaisheshika school of philosophy accepts inferences based on prior knowledge, but also accepts as true the testimony of reliable experts (shabda), thus invalidating logic as a means to truth (truth in the western sense). https://en.wikipedia.org/wiki/Pramana#Vaisheshika_school
3. The Mohist philosophy of China (from the followers of Confucius) admits analogy as a form of knowledge, but rejects formal logic in the western way.
4. Zen Buddhism often castigates logic as not only useless but potentially harmful in one's quest to arrive at truth: http://ccbs.ntu.edu.tw/FULLTEXT/JR-JOCP/jc26600.htm
Your idealism of 'logic' (and given that you didn't even specify which logic, I'm taking it mean you think there is only one, and that it is right) is, when considering the grand arc of human history, indefensible. You argue that it is 'right' and the way to the 'truth', but you can't even fathom that many societies, cultures, groups, and philosophers did not think it useful or necessary or valid.
Simply believing logic is right, without any consideration for an alternative, amounts to a religious belief. Indeed, given that logic remains popular mainly due to the theological musings of Thomas Aquinas, it is moreover, a heavily Christian (from Aquinas) or Muslim (from the Muslim theologists studying Aristotle) belief.