Why? The stories that were written down were based on oral traditions, which continued to exist as oral traditions among the common people, who were generally illiterate. Abraham probably received the oral tradition from his parents, took it with him to Caanan, and passed it down to his descendents, who eventually wrote it down after the Kindgdom was established.
If Abraham brought the writings with him from Messepotamia, then there should be some archeological evidence of Summerian writing in Caanan from the time before the Israelites went to Egypt. In fact, we don't.
Moses had some really good sources for the work he set out to do. I suggest he had written documents.
Remember, Sumerian is a stylized, but still hieroglyphic technique of record making.
It is believed by many specialists that the Sumerian method, in its earliest forms which were more clearly pictoral, were adopted by the people who created Egyptian hieroglyphics.
These same folks also appear to have invented Chinese writing.