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The Babylonians used a "base 60" counting system. There are a variety of reasons given for that choice ~ but no one really knows why.

A more ancient structure relies on a "base 12" counting system. You can derive 12 groups of 1, 6 groups of 2, 3 groups of 4, and 4 groups of 3, and 2 groups of 6.

You can also subdivide the structure into a "base 10" count of 2 groups of 5, 10 groups of 1, 5 groups of 2 ~ and all of it with a "bracket".

You can add "closed fist", "open fist", and so forth, and combine all of that with a three tiered totemic structure. The premier layer are the animals ~ each with basic meaning associated with common elements of life in the Paleolithic. The second layer could very well be assigned to various human virtues, and a third layer associated with natural virtues/events (storms, lightning, etc.).

Simply pairing a number with a totemic symbol can yield a very large number of symbols ~

Such systems are known. They haven't all been worked out but they're out there, and they are very old.

Any good explanation of the development of the Sumerian hieroglyphics invariably harkens back to the association of counters with a simple picture of the items being counted.

You could have 27 associated with a cow, or 38 associated with a bull ~ and that would mean about all you'd need in a society of pastoralists (like the folks who founded civilization in Mesopotamia). Once they had sheep, goats, flocks of ducks, chickens and so on, you'd have more characters, and finally, if trade advanced to tools and ceramic or clay products, there'd be even more characters.

Or, as is more likely, the scribes and accountants of the earlier pastoral groups knew another much more ancient system and simply selected out of it "ideas" that were of use to them as they trekked around the countryside chasing cows. The good stuff was left behind at the tribal graveyards.

Relating "letters" to "sounds" probably didn't work out well before 2000 BC simply because travel was too difficult, distances too far, divisions among related linguistic groups too easy. However, once an "alphabet" became feasible it popped up just like earlier hieroglyphic systems, and before them ideographs, and grave marking systems.