I don't see why they couldn't take Edith to the future with them. Same result, so peace movement. Kirk and her live happily ever after in the 23rd century.
This has been brought up before. The Guardian made it clear that time resuming its shape depended upon Edith dying.
But think about this. 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, whereas in Kirk's timeline that bum was actually reformed and went on to make a substantive contribution to history? You can't pluck Edith out of the timeline without risking other consequences.
The episode already gave us a standard by which someone could disappear from the timeline without consequence: the bum who mugged McCoy and then accidentally phasered himself away. Edith doesn't fall into that category of someone at a dead end of history, so she can't just be plucked out without consequence.
: Too bad you were disappointed.