In my experience, if the kids are well behaved, they are welcome anywhere. I used to work with the technology they use for Le Petit Chef but have never wanted to try that. But maybe I will on our next cruise. I've heard mixed reviews about the food quality.
On the Eclipse for the Daniel Boulud dinner there were three kids with their parents. The kids were next to us and seemed a bit bored. I'm not sure if that was before or after I told them my life story. They seemed to like the food well enough.