Personally, I think the story of Anastasia's scar is just a myth that grew out of the Anna Anderson case. Photos of the girls as toddlers show Anastasia wearing fringe well before she turned three. These, for example:

In fact, all the girls with the exception of Olga had bangs/fringe as toddlers. You can see it on Tatiana in the 1899 formals; by the 1901 session it had begun to grown out. Maria's fringe is visible in the 1901 session, as well as informal shots from 1900-02 (see above). IMO, the question is not so much why Anastasia had bangs in the first place, but why she
retained them.
Here's my best guess:
It's obvious from the 1910 formals that Anastasia's hair had considerably less volume than her sisters'. Perhaps it was deemed easier to trim some of her front hair into a fringe than to tease it up into the fashionable
Gibson Girl poof. (I suspect Tatiana had the same problem, and that she used a rat to correct it -- compared to Olga and Maria's, her hair is strikingly droopy in the 1910 formals.)