Yes, ancient literacy rates were extremely low, but it only takes ONE GUY to be literate enough (and what's even more rare, literarily minded enough) to put down some words for posterity's sake. If there were a million people in and around Israel during the early church days, that's a million chances for one such person to happen along. And in fact - of course - we got at least four. But how early? I say the odds are equally as good that at least ONE GUY managed to record some source material during Jesus' ministry itself! (I posted 15 times this year on who I think that person probably was, but that's beside the point at the moment.)
Yes, communities of people pass on stories by oral tradition, but writing isn't done by "communities". Writing is a solitary endeavor. The writings of ONE GUY could find an audience (especially if his pages were from God) because Jewish communities valued writings. A reader stood at every Synagogue meeting to exclaim the Hebrew scriptures. Parents would tell children stories about Abraham, Moses and David, but they also took those children to hear those stories read out loud, every Sabbath. Of course, each passage they heard read had been written by ONE GUY.
If there was ONE GUY who could write (who had practical and/or professional reasons to think about writing) he'd have noticed elements of writing in the Synagogue readings. On certain days, the reader would have explicitly reminded such a writer that God told Moses to write. God told Isaiah to write. God told Jeremiah and other prophets to write. And surely God had something to do with all those other Hebrews who wrote down the rest of the Scriptures, too.
Now, if you were that ONE GUY with some practical or professional reason to think about writing on a regular basis, and if you were a follower of Jesus... wouldn't you think Jesus' story and sayings deserved to be written down and read aloud as much or more than the writings of Moses and the Prophets? And wouldn't you get started right away... or would you suddenly become like a common man, like a non-writer? Would you just talk about it for decades like everyone else, recording a final memoir sometime before you died?
If you were that [rare but probable] WRITER, I daresay you'd write. Right away.
There are bad writers and there are good writers, but both kinds just can't keep from writing. All bibliobloggers should know this! ;)