A core part of many abusive relationships is the message "No one else cares about you, no one loves you, I'm the only one who will help you". The abuser will socially isolate the victim, cutting off friendships & family. What friends & family remain, the abuser will work to discredit or estrange, all the while telling the victim the abuser is the only person they can rely on.

I think you are imagining abuse victims as fully rational actors straining to break free, if only they had somewhere to go or full custody of their child. But a long time victim of this kind of abuse has been taught they are helpless, worthless, and alone. The only thing more frightening than staying, is leaving.