We also know that groups without writing systems were historically able to adopt writing systems rather quickly, which is, I think, rather good evidence that writing is a technology, not a point of speciation.
Going back to Ancient Greece, Socrates didn't even believe in the effectiveness of writing for communication of knowledge. My poetry professor used to spend some time on this, because it's intimately tied to the art of poetry. He would cite a number of studies showing our emotional responses are intimately tied to our language processing, and that humans are wired to emotionally respond to, and remember, stories.
Even before writing, oral histories were passed down for many generations. For an extreme example, see: https://www.sapiens.org/language/oral-tradition/ .
I won't pretend to know what makes us human, but ultimately I believe it has to be rooted in something neurological, not technological.
In terms of intelligence, yes, but in terms of what we've achieved then "technology" such as writing/archiving certainly has made a massive difference, else we'd be limited to what could be built by passing down oral history and skills passed from one generation to the next, much like Aboriginal Australians.
I suspect that the neurological (& vocal) differences that make us more intelligent than other apes are likely extremely few - more like "fine tuning" differences than anything major.
I'm not sure I understand why this would necessarily be what you would expect. I would say it's entirely the other way around. If we have some sort of favorable evolutionary circumstances that predispose us to turn into archivists, that might be at the frontier of the outer limits of the capability we're able to reach, so it might only show up in certain pockets or subsets of our overall population. Getting there would still hinge on favorable probabilities and circumstances that might only obtain in a small percentage of cases.
As for Socrates, I must confess I am rather smitten with him as a historical figure and as a philosopher, but for the many great things that Socrates is, I don't think he's a reliable authority for the evolutionary history of humanity writ large. I suspect that you're entirely right that oral traditions are more emotionally resonant and powerful than written traditions. But don't think there's any logical fallacy or contradiction in supposing that nevertheless a written tradition could emerge in parallel with oral traditions.
I suppose I do agree with your end point though, which is that I'm not sure that a disposition towards the writing can be pointed to as like a singular thing that's at the essence of what it is to be human. In fact, I would say that that very question is kind of romanticized and abstract in a way that doesn't make clear contact with our scientific understanding and therefore is kind of a malformed question. But I don't have to agree with that form of question to nevertheless believe that our capability to put language into a written form had rather transcendent consequences for us as a species.
As I mentioned in another comment, I'm skeptical of the questions that imply a kind of species essentialism, suggesting that there's such a thing as a one particular trait that distinctly makes us human. I think the real answer to questions like that are vast convergences of immense clusters of facts relating to our evolutionary history and our morphology and so on. I don't think there's any like one single thing. But I do think in comparison to other species a rather elegant way of distinguishing this is to put to our written traditions which as far as I know don't really have any precedent. And if that doesn't blow you away in terms of how miraculous and special are evolutionary trajectory is, I don't suspect anything would. But the important thing is that you don't need a species essentialism to be impressed with who and what we are.
Intelligence - predicting the future rather than reacting to the present - unlocks the possibility of communicating about the future rather than just the present (basic animal calls - predator alerts, intimidation threats, mating calls, etc), which means the message can have value in the future if stored and transmitted (unlike a predator alert which is useless if not delivered in the moment).
It seems that writing, or proto-writing (drawings become symbolic glyphs?), probably developed before message carrying/sending, which then becomes the big capability unlock - the ability to send/spread information and therefore for humans to become a "collective intelligence" able to build upon each other's discoveries.
It's interesting why some groups of humans never culturally developed along this path though - aboriginees and forest peoples who have no written language. Is it because of their mode of life, or population density perhaps? Cultural isolation? Why have these groups not found the utility for written language?
Apparently as recently as 1800 global literacy rate was only around 10% - perhaps just a reflection that in the modern world you can passively benefit from the products of our collective intelligence without yourself being part of the exchange, or perhaps a reflection that desire for information is not the norm for our species - more for the intelligentsia?!
It fits the basic concept of writing, where complex ideas are communicating through time rather than space.
As to using symbols it depends on what you mean by symbols. Though an ants nest is a dark environment so writing the way we think of it via coloration would be a useless. In such an environment pheromones have inherent advantages, but you can’t get the same highly detailed shapes.
Given that with exception of an interesting knot-encoded (Quipu) system for some period of the Inca empire, the entire human population of the Americas (at least 15-20% of the total human population) fits this description, the answer seems to be "no". These cultures built huge cities (among the largest in the world at the time), used sophisticated irrigation schemes, ceramics technology and more.
It's impressive, and perhaps a bit surprising, what a culture like the Incas were able to achieve using only (we assume) oral tradition and passing down of skills, but there must be a limit to the complexity of what can be passed down this way - more along the lines of the skillset of an individual man being passed to an apprentice, but presumably also "managers/planners" passing down skills as well as tradesmen.
The Maya though absolutely had "full writing" at least during the classical phase around 200 - 800 AD, and their writing probably derived from the older Olmec system. The Maya system worked much like classical Japanese or ancient Egyptian, and uses a mix of ideographs and purely phonetic symbols. They could, and did, write anything they could say. There are historical accounts of battles and names and etc. given in monumental inscriptions. And it is almost certain during the classical era that the Maya had full literature, with stories, mythology, histories, anthologies, religious scripture, etc. There were probably still priests who could read the Maya system when the Spanish arrived - one Jesuit recorded notes that indicate some of the phonetic values of some of the symbols correctly! - but they were mostly killed and most of the codices burned by the conquistadors.
It was not deciphered again until the 1970s and it's sadly still not widely appreciated that Mesoamerica had literate civilizations.
Fun fact: The Maya were such good astronomers and record-keepers that we have exact date alignment, which we have neither for Ancient Egypt or for Mesopotamia. For example any dates in Assyrian records are subject to a ~40 year window of uncertainty for exactly when they occurred. There is no such ambiguity for the Maya: Pacal the Great almost certainly died on August 29, 683.