There really lacks logic in this why the “real father” appears only after the information is revealed to the whole country, or wouldn’t it be more logical to do the blackmailing when no one had made this information public yet?
And I also find it hard to believe that for so many years “maybe the real mother” would hide her love for her biological children (unless it’s some other relative).
Now a small theory the Father was aggressive, evil, maybe a drunk, and the mother, knowing the Shiunji family, asked Kaname and his wife for help (and maybe after the court case she won I added this to make it make sense and be in accordance with the law because I doubt that simply running away with the children would be enough) they accepted her into the family and also declared to raise the children.
I don’t know, this plot is slowly becoming more and more stupid.