So Sybille was already married as well when she started her affaire with Albert? oh boy, their relationship didn't tear one family but two apart, both were selfish people who only thought about themselves.
Delphine might get legal recognition and of course will get a share of his state once Albert dies but she will never be his daughter, because he doesn't want to acknowledge her as such, if she was only after being a part of his family, she wouldn't have gone to these extremes, maybe she would have started reaching out to her siblings (if she has a relationship with Laurent that would have been a good starting point for example) and then slowly work her way into the family, maybe Albert would have been more open to work on a relationship with her that way.
You can't force someone to love you and accept you as a part of his family, on paper she will be a daughter of his, but on his heart and his mind, she will be nothing but a nuisance, someone who's trying to destroy his brother's and son's legacy and someone who's attacking the monarchy.
Albert has been a complete douche in this issue but it has been out of his own choice (first forced by Baudouin no doubt but then he chose to get back with Paola), Sybille was another willing participant who didn't think about the consequences of her acts and is now trying to get into the limelight.
Also, has Delphine seen how her legitimate siblings turned out? Phiilippe and Astrid were virtually raised by Baudouin and Fabiola, Laurent was just cast aside most of the time and Albert was always a terrible father, if you can call him that because a man who behaves like he did is hardly someone you want to have by your side. Both the King and his sister don't look close to their parents at ALL. Does she really want to have her own children close to a man that has been that horrible to her and her siblings?
I really really really don't think Delphine wants to play happy families now, I think she wants money and she wants revenge, Albert brought it on himself by making tons of mistakes when he was younger but it doesn't make her actions look any better, although this might prove that she is his child, both go for what they want, without thinking of consequences and without thinking about anyone else who might stand in the way.
There are many ways on which this issue could be worked but Albert refuses to budge and Delphine refuses to work on a low profile answer to this (I think Philippe would have agreed to work on this with them to avoid further scandal but both are very stubborn).
I wonder how Delphine's kids feel about this? would they like to be dragged around the media thanks to their mother's wishes? how about Philippe and Astrid's children?
Massive massive mess that started when two people decided to break their marital vows to pursuit the thrill of an adventure, something that snowballed out of control and keeps causing grief to all involved and many collateral damage to those who are close to either party but have nothing to do with the choices made.