It doesn't necessarily mean that the story was stable, it's just the version that got to us.
There are serious attempts made to write down the vedas. The thing is, historically, very few people learned all 4 vedas by heart; Instead different families recited very small part and passed the recitation as heirloom.
If you meet all those families and compile their recitation, it exactly matches to what we have from different earlier efforts of canonisation.