It always rather gets on my nerves when people declare democracy to have begun in Greece. As I understand it, democracy in Europe developed mostly in the modern period with a gradual political enfranchisement of more and more of the population as royal power waned, and there is virtually no meaningful continuity with the strange, exclusive and electionless system of ancient Athens. That often causes me to doubt how much western culture truly "began" with Greece and how muh of that notion is simply renaissance and enlightenment Europeans fetishising an idealised classical aesthetic.
Still, it must be said that the field of philosophy in particular seems to owe much to the Greeks. Pretty much everyone has heard of Plato and Socrates and Epicurus. Indeed, one of the first images that pops into one's head when the term "philosophy" (a Greek one no less) is mentioned is of old Athenian men sitting around and musing on the nature of things.
Why is this? Were Greeks simply exceptional? Did they write things down more? Did the structure of their society allow for learned men to spend their lives pondering things, more than other societies did?
Or did most great civilisations produce such thinkers, and we simply have disproportionate evidence from classical Greece? It's worth noting for example, that the works of ancient Greek writers of all fields, from Herodotus to Plato, are most often preserved in Medieval European (or sometimes Arabic) copies, without which we would not know much of them. It seems unlikely that Greeks had some kind of inborn intellectual talent, especially when we consider that other civilisations produced marvels that rivalled or surpassed those of Greece (sometimes centuries or even millennia earlier) that must have required a society of great intellectual capability to produce.
Were there, then, scientists, philosophers, historians and the like in ancient Egypt, Mesopotamia, Persia etc. whose works have simply not survived? Is it plausible that the equals of Socrates dwelt across the ancient world but have left no traces? An Etruscan Plato, a Persian Epicurus, Mesopotamian Thucydides and Egyptian Aristotle? Surely you'd expect them to if they did? If so, why do we not know about them, and if not, why did Greece alone seem to produce such men?
As a secondary question, was Greece perceived as intellectually unique at the time the same way it is now? Did the Greeks consider their own societies the pinnacle human thinking, and did their neighbours view them as such, or is this a modern phenomenon?
There are a lot of questions here, and it is not feasible for me to answer all of them in one comment. There is also the fact that some of your questions are phrased such that they are, by their nature, unanswerable. (For instance, "Is it plausible that the equals of Socrates dwelt across the ancient world but have left no traces?" -- There's no way to answer a question like that. By definition, someone who left no traces is someone whose existence is totally unverifiable.)
I'll have to tackle just some of your questions, then.
did most great civilisations produce such thinkers, and we simply have disproportionate evidence from classical Greece? It's worth noting for example, that the works of ancient Greek writers of all fields, from Herodotus to Plato, are most often preserved in Medieval European (or sometimes Arabic) copies, without which we would not know much of them.
(Consider, firstly, that we often define a great civilization as one that does produce such thinkers, making the answer to the first part of this question a trivial 'yes'. But I don't think that's exactly the spirit of your question.)
This misses the connection between the greatness of the Greeks and the survival of their texts. The preservation of manuscripts is no doubt partly coincidental (after all, an accidental fire might make the difference between a text surviving to this day and not surviving), but for thousands of years, Plato's texts, and others, were preserved precisely because of their importance. Nobody would have wanted to preserve his texts in the first place if they were not singled out as exceptional and important -- and the same goes for Epicurus' letters, Thucydides, Aristotle, etc. This does not mean that someone whose works are lost to us (e.g., Chrysippus) is unimportant but survival of manuscripts does indicate importance.
You might like these r/askhistorians threads:
https://www.reddit.com/r/AskHistorians/comments/16pk82/how_did_ancient_greek_and_roman_literature/
https://www.reddit.com/r/AskHistorians/comments/57f8a0/why_do_historical_texts_become_lost/
https://www.reddit.com/r/AskHistorians/comments/1av18t/how_did_sources_from_antiquity_survive/
Were Greeks simply exceptional? Did they write things down more? Did the structure of their society allow for learned men to spend their lives pondering things, more than other societies did?
You get at something important when you ask about writing. The ancient Greeks definitely tended to write things down more than other cultures. Especially as the history of ancient Greece proceeds from the archaic to the classical period, there is a gradual increase in literacy rates and a corresponding increase in the texts that we have manuscripts of today. We hardly have any philosophy from the archaic period and even from the early classical period, for instance, except for fragments. But we have Plato's corpus in full. There is no doubt that North America's Indigenous peoples, for example, had large, sophisticated philosophical traditions, but since it was communicated basically entirely in oral traditions until very recently, there just are no Indigenous philosophical texts to suggest as reading in Intro to Philosophy classes, for example, in the same way that you can recommend anything from basically any part of Europe's history in the past few thousand years. (Mind you, you could, of course, assign contemporary Indigenous philosophy.)
And of course there are rich Indian and Chinese philosophical traditions: Buddhism, Confucianism, Taoism, and so on. Just to name a few. These traditions go back as far as Plato and sometimes even farther.
And yes, ancient Greek society did tend to lend itself to philosophizing. Plato was rich and never had to work. But that doesn't explain very much: many people have leisure time but don't spend it producing amazing works of philosophy. And Epictetus, a Roman philosopher, was a slave.
But does this mean that the Greeks were not exceptional? Well, that really depends on what you think makes someone exceptional.
If you think that having a society that frees you up to produce exceptional things doesn't make you exceptional, then the ancient Greeks were not exceptional.
If North America, China, India, etc., having rich philosophical traditions makes you not exceptional, then the ancient Greeks were not exceptional.
But that doesn't really get at what people are talking about when they talk about the ancient Greeks being exceptional.
What people mean is that Plato, Aristotle, etc., got at some very important insights about what it means to be a human being, and while other cultures did too, the ancient Greeks produced achievements that were unlike any other civilization's. Specifically, the ancient Greeks invented concepts, distinctions, theories, ideas, etc., that form the bedrock of our own conceptions of ourselves. Many of the ways we think of ourselves come directly and in observably continuous ways from ancient Greeks. Virtually every idea that Christianity promulgated has been inflected by Plato, Aristotle, and Plotinus.
This also brings us to another sense of 'exceptional': people generally think that, in some way, the ancient Greeks got it right. On this view, China, India, etc., might have complicated philosophical systems but they are wrong. Sure, Aristotle, Plato, etc. are not right about everything (we can talk about ancient science being wrong, or maybe even their political views getting things right or wrong), but there are key ideas there that they got right and everyone else who has gotten it right has just taken a page from their book. (Think of Alfred North Whitehead saying that the "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato.") People who did not even take a page from their book have gotten it wrong in some significant, relevant way -- that's another way we might understand their claim to exceptional status.
I feel like my response just generally scratches the surface of your questions, but I think that's the best I can do right now with such a large, full post.