That a society has writing (usually confined to a small class of scribes before the invention of easy-to-learn scripts like the alphabet) doesn't necessarily mean that anyone is composing historical narratives. The Greek alphabet dates to at least the 8th century BC but the first Greek historians wrote in the 5th century. That said, there may have been various earlier written texts for an Israelite historian to have used--in fact in the Bible there are references to now-lost earlier books.
“the ancient Israelites were recording their history in real time as opposed to writing it down several hundred years later.”
Is anyone making the argument that the Old Testament is inaccurate because we had no evidence of Hebrew writing going back this far?
Even if we have evidence of it, it’s not evidence that the bible was written in this period. Nor, is it evidence that the bible wasn’t written in this period either. We simply don’t know the exact period of composition of all the books in the Old Testament. And, not all of them were written at the same time. Genesis was old even in the time of Moses.