I’m sorry I can’t answer you, because I don’t know. But I love the question and have several more myself.

Do we know every invention of writing or just the surviving ones? What is writing? Are hieroglyphs writing? Where do hieroglyphs end and cave paintings start?

Two relevant tidbits of info:

1. When Sequoyah created the Cherokee writing system, he had to prove its usefulness to the Cherokee elders, in order to promote its adoption.
2. Plenty societies had access to writing, but it was only used by a ruling elite and/or a cultural elite, with most of the population being illiterate.

Based on those two things, I think that the usefulness of writing only becomes useful for most people if they know how to write or have heavy contact with the ones that do it. I mean, you can remember most stuff just fine, and you can speak with other people, so what's the value of learning a weird system to put those words in material form? So it's one of those things that only appear rarely but, once it appears, it spreads fast.

It would be useful to list and analyse those few cases where writing was invented, i.e. were there any particular circumstances that were especially conductive to creating a writing system that weren't present elsewhere.

My guess would be that trade and territorial spread of the given state are very useful for inventing it, since it's needed to calculate and store data, to communicate across greater distances (sending messages to other towns that you trade or have some relationships with - not necessary in tightly-knit tribal communities)...

And once someone invents writing, which is a pretty difficult thing to do (especially to teach it to others and make it actually durable), it's obviously much easier for anyone who comes into contact with that culture (which is likely to happen if the culture trades a lot or covers a large territory) to just imitate and adapt their writing system rather than invent everything from the ground up.

This is ofc just my theory based on what I know about the Near East (e.g. Phoenician alphabet > Greek > Latin & Cyrillic).