So, it’s not been rewritten, but rather one person said that that was her age and people hundreds of years later accepted it as true.

To see how you need to understand the nature of Hadith. It is a massively distributed system. No one person really has control of the narrative. Anyone could repeat something they heard and people would accept it as true if it traced back to a respectable source.

In the case of Aisha’s age, the young age traces back to the great grandson of her sister.

Why did he say it? Who knows. I don’t think he was lying. He said he heard his dad say it. Some say his memory was poor in old age.