Responses
I think it's because that's not her priority after being reborn. Repairing her relationship with her family isn't available because it's not a good one from the very start. You can't repair what isn't broken. She's still a child from forced marriage. So rather than solving the problem (relationship to count family), she eliminate the problem (leave the clan).
Her being cold I think it's because of her hardship. Whatever blinds her heart in the past still leave reminiscence, but I think this can progress.
The start was a little hard to follow at first but it cleaned up so I figured it out. But the one thing I find odd is why our leading lady didn't change more of her reborn past. That and why is it that even with the changes she is so cold? I'm sure we'll learn more but there is so much confusing information.
What I know so far is that after the girl was born she was ignored by her mother, tricked in to killing the first wife, thus being abused by her Father, half brother, mother, and her maid was killed. She killed the man she believed was her Grandfather, but her mother appeared to not vanish. She went in to sword training, met the black King who clearly wanted to what appears to be marry her, but he became full of mad hate for her and she died.
Then she was returned to the past. She was still ignored by her mother, the first wife lived, she was abused only by certain maids but only ignored by her Father which he regrets, she began sword training young, her maid lived, her mother goes mad, she kills both her mother and the man she thought was her grandfather, learns she has the power of a former Goddess and that her mother was one once, and now she's returned to the point where she will meet the King.