@Luna
everything went wrong because the mother lied. Long story short, the mother decided "let's lie, which will cause my daughter to have a horrible life, because that way my lover won't feel so sad over my death". If she didn't make up that lie that she hates the father, the father would've been sad over her death but he would've raised FL and possibly even gotten over the mother's death faster by focusing on loving their daughter.
Mostly it's the author's fault. The story needed a misunderstanding for the drama and they wrote it in the cheapest way possible. That's why no one in the story is directly blaming the mother but if you look at it from an objective point of view, yeah, the mother is the cause of literally everything.