Native Americans (or as they are known in Canada, First Nation Peoples) colonized the Americas thousands of years ago, and estimates vary, because carbon dating for the time is very innacurate.  And it could be that the native Americans developed the sling once they arrived in America, as it's use is not known to be universal.  The atlatl and bow were developed indepently of everywhere else, the atlatl inumerable years ago, the bow fairly recently, around 100 BCE to 500 CE, dependent on where you are thinking of.  So the native Americans can only give us a rough estimate, or not even that.

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It is a good assumption for the discovery date of the sling, although the absence of archaeological specimens don´t implies the nonexistence of the weapon, been perishable its materials. The fact that it don´t exist archaeological projectiles of sling in Australia, could mean that it was not longer used in the times in which manufactured projectiles were used, but not that the first settlers did not take the sling with them, who would use it with natural stones that can´t be identified. It is possible that the sling was not useful in the Australian environment, or who  knows. And also, as you say, it is possible that first Australian settlers did not take the sling with them, but that doesn´t means that  it  didn´t exist previously in other places.

The well-known writer Jean M. Auel, authoress of the prehistorical saga of Ayla (so admired by Ulrica) place the use of the sling in the time in which Neanthertals and Homo Sapiens coexisted, that is to say, around 40.000 years B.C. Of course we can´t grant scientific value to the viewpoint of a novelist who is not antrhopologist, but she certainly counted on the advising of archaeologist and anthropologist to write his novels.
Nevertheless, the archaeological evidences place the appearance of the "bolas" long before, around 200.000 years B.C. when other previous human species, the Homo Erectus, had spread from Africa to everywhere. If we admitted that the sling could be derived from the bolas, it would locate its discovery long before the  40.000 years. But this is only a possibility, the invention possibility of the sling, because its archaeological evidence  does not appear until the Neolithic, around the year 7.000 B.C., date in which are dated the clay projectiles.

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Yes, the thing about the bow is true.  And atlatls are/were used literally the world over.  Every single culture in the world is thought to have used atlatls at some time.  In Australia, aborigines still use them (they call them woomeras), Inuits use them, and they use the phrase throwing board, rather than atlatl.  Atlatl is a nahuatl word, the Aztec language, which is why many people believe the atlatl to only have existed in the Americas.   The oldest atlatls in the world are from north west Africa, dated at 25,000 years old, although many believe that atlatl use was widespread in Europe and Africa around 40,000 years ago.
What I meant was that the atlatl was developed by most native American groups without influence from elsewhere, despite the fact that everyone was using them.  The atlatl is the weapon I like the best, especially my newest one, a Tlingit style atlatl.  Looks nice as well as practical.  Anyways....