It's likely that there have been bottlenecks, where a single written version became the main common ancestor to copy from. Long after the oral tradition died down and other written versions were lost. Or because some patron decided to fund the dissemination of a particular copy, like Guttemberg or King James, or the Toledo School of Translators. Or because a particular heir of the oral tradition wrote it down, like Homer.

It doesn't necessarily mean that the story was stable, it's just the version that got to us.