I am sorry you are hurting. I speak from the other side, and hope that my perspective is helpful to you (as yours is for me).
When a grandparent is overly eager to bond with a grandchild, they often make the mistakes of neglecting their relationship with the parents and overstepping boundaries. This is further complicated if the grandparent harbors a false sense of entitlement in regards to their rights with the child. They feel they are owed a certain amount of time, equal treatment to someone else, or are excused from breaking normal social boundaries with the child simply because they are the grandparent.
When I read your post, it sounds like you are very eager to have a close relationship with your granddaughter, but that you have no interest in having a close relationship with your DIL, or even your own son. It sounds like you view your DIL as merely the gateway to your granddaughter. I'm sure that your DIL senses that you want her to go away so that you can have her child all to yourself. This is a recipe for her to feel threatened. You write that you just had to watch your DIL play with your granddaughter last visit. Perhaps your DIL is trying to tell you that she would like to be included in these visits, rather than you monopolizing your granddaughter's attention and excluding DIL? When you visit, you should be equally interested in visiting with DIL, as she is also your family member. If not, she will feel like you are competing against her rather than joining her in loving the child.
If DIL's parents are more involved than you, it is because they have a stronger relationship with your son and DIL. If you change your attitude from seeking your (imagined) rightful share of your granddaughter, to trying to be supportive to the family as a whole, you will likely find yourself included more.
It is not appropriate to take someone else's child on a walk without the parents' enthusiastic blessing. You could maybe offer if the parents seem like they need a break. But you should never just walk off. And if you took the child away during a group gathering, it probably seemed like you just wanted her all to yourself. This behavior likely came across as possessive and boundary-stomping to your DIL. She likely fibbed that she was sick rather than tell you to stop monopolizing her child.
In summary, the more your push to bond with the child while putting little effort in to developing a relationship with your DIL, the worse the problem will get. If you want your DIL to do more than tolerate you, then you have to treat her as more than merely the gateway to your grandchild. Your relationship with your granddaughter will come naturally if you have a good relationship with both of her parents.
