Without witnesses to her death and a body, how do you know that one of the bums at the mission wouldn't be blamed for her disappearance.
So just make it openly known that she is "running away with" that handsome new boyfriend of her's that everyone has seen her with, the one she told all her girlfriends about.
And if the Guardian wouldn't move Edith into the future, Kirk could actually take Edith away somewhere out of the way in 1930 (because he loves her) and just quietly live out their lives in a small town.
Screw Spock and McCoy by the way, he's in love.
And what if Edith doesn't agree to run away? And even if she does, how is the timeline assuredly restored?
Spock's tricorder recorded only two histories. The one in which she dies in a traffic accident is the history that leads to the Federation. The one in which she lives is the one to be avoided.
The parameters of the story didn't allow any other option. The unpalatability of the necessity of her death is the point of the tragedy.