But... IDK if this (or other clearly advanced writing systems) demonstrate "refinement of millennia."
I think we have a "history is accelerating" bias. Changes in the deep past happened slowly, and the pace of change increases over time. That may be true from a very broad POV... but I don't think it's true on shorter timescales.
There are no hard limitations on going from a newly invented writing system to a professional scribal culture in a single generation. I don't think watershed "revolutions" are something new. Egyptian writing, and Early bronze age egyptian culture more broadly gets very advanced, very quickly. We don't really know what elements have deep histories... but it's hard to explain ancient egypt without allowing for some impressive leaps. Hence aliens.
Also... "common ancestor" can be a lot of things. It could be like the gradual species-like philogenetics of cyrillic, latin, hebrew, arabic and all other alphabets' development from proto-sinaitic and canaanite/punic. The same script gradually evolving in different scriptural islands.
Otoh... "ancestry" can be pure inspiration. The idea of writing, its uses and the certainty that widespread literacy is possible can be the "dna."
The confusing part is that culture does, often, evolve very gradually like species and clades over time. These sometimes leave evidence of the whole process. Sudden explosions can't be deduced from the absence of evidence.
We won't, by definition there's no written record pre writing.
It sucks how many instances in the historical record are like "welp, they had settlements that point to thousands of people, they made pots and they buried their deceased" that's kinda all we know about places that stood for millenia.
Even written records of oral history of the invention of writing would work
The bigger issue might be that "the invention of writing" is a rather boring and gradual event. Some administrator starts tracking the state of their grain stores with some symbols, then they start tracking other things and need symbols to differentiate, and over time more and more meaning is encoded in those symbols, until we call it writing
How can you tell that a script is "refined", especially from a single example?
The refined scripts typically use fewer and simpler symbols. The only exceptions to this tendency towards standardization and simplification are in the case of some script variants whose main purpose is to be decorative, not practical, e.g. which are intended for inscriptions on monuments.
单子是自函子范畴中的幺半群
It bears remembering that spoken language existed long before written language, and written language developed as a form of encoding spoken language. Purely pictorial communication utilises a small number of large symbols that make it clear what is being conveyed from pictures alone, but the language depicted is too complex and abstracted to be purely pictorial; it uses a great number of small symbols, and you cannot understand what it is trying to convey merely by looking at it as a series of pictures. For a reader to understand what is written there would require understanding the relation of symbols to spoken language.
The main issue with hieroglyphics is that it was a very convoluted system, where symbols meant different things depending on context. A symbol of a bird could be a literal bird, some abstract concept related to that bird or part of the sounds of the word for that bird.
In middle egyptian (the language you probably assume) "pictures" are just syllables. They are phonetic, not semantic, in the same way letter of modern language correspond to sounds, not meanings.
Egyptians had no problem expressinyg conplex concepts and they also had cursive writing, which is much easier to write.
The phonetic symbols included in the Egyptian writing system represented 1 consonant or 2 consonants or 3 consonants, not syllables. Any syllables or short syllable sequences with the same consonants were written with the same symbol.
This makes the Egyptian writing system an exception, as all other writing systems that have developed completely independently, instead of being inspired by an existing system, have used phonetic symbols for syllables.
This is the very reason why the Egyptian writing system has generated the ancient Semitic alphabet with 29 consonannts, from which all later Semitic consonantic alphabets have been derived, then the Greek alphabet and other European alphabets, and the Indian writing systems and other Asian writing systems derived from them.
Since the beginning, the Egyptian writing system had two variants, depending on the writing instruments: hieroglyphic for inscriptions carved in stone and hieratic for texts written with a reed brush on papyrus. The latter is what you mean by "cursive". "Cursive" is not really appropriate, as hieratic was still a very complex script, difficult to write, even if it was simplified in comparison with hieroglyphic. Millennia later, a more cursive form of hieratic developed into the demotic script.
The point stands still: the writing was not as clean as modern alphabets but was capable of expressing abstract concepts, it is completly orthogonal to concepts expressed in writing.
I'd wager the Great Pyramids will still be around in 1000 years and the Burj Khalifa will not, if anyone wants to take bets.
The closest analogy might be: modern alebraic notation is compact and clean but this doesnt mean algebra didnt exist much, much earlier.
Using the phonetic sign subsets of the Egyptian and Sumerian scripts, it was possible to write any sentence that could be spoken in their languages.
This was the most important advance in writing and both in Mesopotamia and in Egypt there is evidence about this transition from an earlier writing system that could write only a subset of the words of a language, so it could not be used to write arbitrary sentences, but only things like lists of objects with their amounts and owners, like needed for accounting, to a writing system that added phonetic symbols for writing any words that did not have their own symbol.
I cannot read the paywalled article, but it seems that now there is evidence that also the Proto-Elamite writing system has also passed around the same time through this transition from having only symbols for certain words to having phonetic symbols too, e.g. for syllables, which can be used to write arbitrary words and sentences.
Before phonetic symbols began to be used, we cannot know the language spoken by the users of a proto-writing system.
While in Egypt there is little doubt that the first users of writing spoke some kind of Old Egyptian, in Mesopotamia there is doubt the users of the first proto-cuneiform writing system spoke Sumerian. However, by the time when phonetic cuneiform signs were introduced, the language of the writers was Sumerian.
In the territory later known as Elam (in the West of present Iran), it is not known what language was spoken by the users of the Proto-Elamite writing system. It could have been an ancestor of the Elamite language spoken a millennium later, or it could have been a completely different language. Elamite is not related to the Indo-European languages that spread much later in that territory, like Old Persian.
By the way, Chinese uses a modern example of such a script and succeeds in representing such concepts.