I don't have experience with this, first off. So my suggestion, though heartfelt, is only from knowing people who grew up like this.
Here's one example. One of my friend's parents were farmers. They worked really hard and didn't have time for their kids. Basically, the grands raised them. They didn't have a lot of supervision either because the grands were pretty old, and the kids probably wore them out. They grew up in a rural area so at least they didn't have issues with gangs or bad influences. The boy was astonishingly smart and got a scholarship to go to college, the first child in any of their families to go to college. He has turned out pretty well, considering his upbringing. He has issues - don't get me wrong. Sometimes I think he is on the Aspergers spectrum - who knows at this point.
Anyway, as far as the boy is concerned (I don't know his sister personally) if it hadn't been for the grands, I think he would have ended up a total mess. I know that your daughter has a "plan" for making him her total responsibility and the new father to take over in his life, but I don't think that's feasible. She is forgetting that he bonded with you as the "other mother", which certainly shouldn't surprise her, yet it has. Instead of her thinking that she's done something wrong, she should see it as doing something right. He has an extended family who loves him and who he loves. And he should be able to keep that in his heart and mind.
What I would suggest is that there are a set number of nights that he can come over and spend the night. Maybe 3 per week at this point, and once he stops pushing it with them, maybe in a year's time, take it down to twice a week. What will be crucial is that you have the same rules and limits in YOUR household as she has for her household. Honestly, if you give him any latitude that she doesn't, you are setting her up as the bad guy and you don't want to do that. You should actually be more strict than his mother. That takes away any incentive for him to try to manipulate you two and play you off of each other.
I would also take the sibling that came into the marriage at least once a week. That way there is not jealousy or a feeling that you only love this boy. That can wreak havoc between siblings and hurt this little family a great deal.
Urge your daughter to accept that he loves you and that it's not a reflection on anything she did wrong. If she is insecure about his love for her, he can feel it and it makes him want the oasis you provide because he feels secure in your love for him. (I hope this makes sense.) The worst thing she could do is try to make him feel bad about it or shame him for wanting to be with you. It would also be very selfish of her. SHE is the adult. He is the child. He has a right to feel this way. She taught him to feel this way for all these years. She isn't thinking right if she thinks he can just suddenly switch gears. It's not possible.
As for my friend and his mother? They get along well, though they no longer live close to each other and can only see each other once a year. He and his father have a harder relationship, but not because the grands raised him. He is divorced from my friend's mother.
I will say that when my grandfather died, my sister and I stayed with my grandmother several nights a week, for at least a year. She ended up selling the house and moving in with her mother, and their place was really small. We visited during the day but didn't spend the night. She needed us, we liked being with her, but it was pretty mutual when we stopped staying every week. It took a while, but that was okay.
Please feel free to read this to her. I hope she will take heart and understand that a loving and close relationship with you is to HER betterment. It makes her son feel more secure, and if she is accepting of it and loving to him, allowing her other children to benefit from your love, her entire household will be happier for it. Continuing to be active in these children's lives is something you should be commended on, if you are always backing your daughter up, not undermining her authority, and keeping at her schedule and rules for the kids. My mother ALWAYS did that. She never said "Well, I'm the grandma and I get to spoil my grandkids." I always appreciated that about my mother...