I don't know if it's the same story you're talking about but I believe it was a 4 year old girl as well, who had a disease that required very painful and extensive treatments and was never going to get better and the disease was going to end up killing her months down the road anyway (and most of that time would be spent in hospital getting the treatments). Her parents let her make the decision to just die at home. It was very controversial but I had no issue with it in theory. The one thing that did bother me was that the parents were telling the daughter all about heaven and how happy she would be there... imo it's too young for that little girl to grasp that heaven is a religious concept and not a fact, because her parents are telling her that it is, essentially. That didn't sit right with me at all. It almost seemed like misleading/tricking her, and not being objective and telling her that hey, being dead might actually mean just being dead. In the video I watched she was so excited about heaven and seeing her parents again, she was looking forward to it. On one hand I can understand wanting to make her inevitable death a little less scary for her, but I also think she deserved to hear it both ways in order to make the decision. I know religious folk will disagree with me, but oh well.
I have a four year old and this would be an incredibly hard decision in a situation where she had a severe illness that was going to take her life, but if she was guaranteed to die in the near future regardless of painful treatments then I would let her stop. Ultimately I wouldn't want her to suffer more than she had to. But if there was even a small chance she would survive with a decent quality of life then I would make her get the treatments, because then it goes from letting her choose whether to die comfortably a little sooner or painfully a little later, to potentially IF she dies, and the latter isn't a decision I could let a young child make.