In Mesopotamia we can see the evolution over many hundreds of years and even a few millennia from an accounting system that used a few symbols for the things that were recorded, to a full fledged writing system that was able to record any spoken sentence.
Before the societies of Mesopotamia and Egypt, which required a complex management of the available resources, there was no need for accounting with written records for great quantities of varied goods.
It is likely that for tens of thousands of years people have been able to make drawings that recorded useful things, like maps, and they probably have used some sets of symbols for various important things, like kinds of humans, kinds of animals, kinds of plants and so on.
Nevertheless, it is very unlikely that any such set of symbols used in the distant past has ever been able to record any complete sentence spoken in their language, with all the grammatical markers that do not have a concrete meaning.
Most human societies that have not seen a writing system at other people have never invented any writing system.
All those that are likely to have invented writing independently (though for some of them it is not certain that they have not seen the writing used by others) have invented writing systems in contexts similar to those of Mesopotamia and Egypt, i.e. in a settled agricultural society where cooperation was required at a large scale and management and accounting of resources was necessary.
However, people generally underestimate how culture differs. The idea of belonging to a certain group was vital in the Middle Ages. And that’s something modern people would probably not put much thought in if time travel was possible.
Single digit thousands of years is a drop in the bucket for evolution. The people back then were largely the same as they are now. They just lived in different cultures. Modern innovations are all motivated by a need. Some form of pressure to come up with a solution. Pre civilization societies just don’t have a need. It’s accounting and management of a large population that requires some form of book keeping and written records. There is no reason to believe, without need and cultural precedence, that writing would have been invented by societies in earlier stages.
In the same way classical physics isn’t wrong about the world compared to modern physics, calling this the first writing system is just in the context of our current understanding of the world of the past. We can be pedantic about the details but going by our current understanding, this is what makes sense. This understanding might change but it’s is needlessly pedantic, especially in a field like this that is less „cold hard truth“-ish compared to physics.
We haven't ever observed the invention of a writing system in the absence of cultural contact with written language, so we actually have no basis in evidence for the motivations that bring such a thing about, we just have speculations based on preconceived notions about why people might use writing, which is fine, but don't pretend like it isn't blindly opining on why you think someone would've invented or used a writing system
The depth of the information contained can be debated, but as the Spanish utilised the related quipu following their invasion, for their own record keeping needs, it is "of use". (The Spanish used those for tracking crop growth, timelines, deliveries, royal decrees, family trees, and spread of illnesses in remote areas. They're also still used in some religious sects today.)
While nobody can prove there wasn't some civilization that had writing prior to Sumer (can't prove a negative), it stretches the imagination to think that a civilization ONLY ever wrote on perishable media and not on, say, pottery or building walls. Then died out without transmitting their invention to successor civilizations, all of which we can account for their development of writing.
Also, what perishable media? The invention of papyrus was contemporary with Sumer's clay tablets.
Art is not writing, true. Distinct symbols and syntax are required for writing. But it need not be transcription of anything spoken.
Yes writing systems evolved from earlier symbols used to indicate general concepts without a linguistic context. And those symbols go far back into the ice age -- we see such symbols on the walls of cave paintings in Europe, for example. In the case of Sumer, we certainly know that the writing system derived from accounting symbols for livestock and grain and such.
But going from simple context-free symbols to transcribing spoken language or thought with grammar and structure is a conceptual leap worth tracking.