When a baby goes missing, LE knows that they MUST "solve" the case. The public demands it. If they can't find clear evidence that the family did NOT cause the disappearance through process of elimination, they begin to focus on the family. Then all gloves are off. They will do anything to imply that the family is guilty. They will lie to the family. They will stalk them. They will misconstrue statements. Literally ANYTHING, until they can piece together enough circumstantial "evidence" to cobble together a workable story. And make no mistake about it - there is enough flexibility in every ones life that a skilled story-weaver could make any of us look suspicious or downright guilty.
Even confessions are not really reliable. I am doing my thesis on false repressed memories. I have seen a lot of data about false confessions during my research. The human brain can easily be manipulated into remembering things that didn't really happen, and when the human is stressed and exhausted it is almost child's play to get a confession - real or otherwise.
This is why lawyers will tell the family to not cooperate. Sure, in the first day or two it is imperative that the family do whatever they can. But at this point, LE is simply walking around looking for a dead body. if the baby is dead, finding her will not change it. If the baby is alive, searching fields and streams will not find her. And the more the family unwittingly cooperates, the more they may be handing LE the hammer to nail themselves, right or wrong.
Now, lets say the family is innocent. They have just had the absolutely worst thing in the entire world happen to them. And NOW, they are being circled by the LE vultures who just need to solve this case. A lawyer will tell them that no matter what they say they will not be able to convince people that they are innocent. And the lawyers are right. Short of finding the baby with DNA evidence otherwise, this family is in for the most horrible ride of their life.
The least we can do at this point is give the family the benefit of the doubt. Look at it as a balance scale. Go ahead and pile stuff on one side or another, but wait until the scale is so far down on one side that there is no doubt before you condemn this family. Just remember, if you are wrong, you are personally adding to the horrific injustice that has befallen this family. If they are guilty, it will come out eventually and your hands will be karmic-ly clean.