I don't think it would be so much worrying about what others would think, as a sense of precious time being wasted and the possibility that LE would take what they find and use it as another reason to hone in on a parent who knows he or she is innocent while the child is still missing.
I think we are all aware that when the investigation starts veering in a certain direction, it often makes a full turn. Then later, if it's revealed that the turn was wrong, evidence is gone and clues have evaporated and the child is gone forever. I want to say that I would do everything and anything LE asked of me, but I think I can see myself protesting if I felt this begin to happen. It wouldn't be so much a lack of cooperation, in my mind, as a serious disagreement regarding the use of resources, and probably some defensiveness if they started to sound accusatory.
Oops, Bessie... didn't see what you said until after I posted... I could have just thanked and not said all of this!
It is probably a good idea to hire a media adviser,especially for anyone who has never had to deal with the media much... But, it cannot be useful to stop communicating with the very people who are charged to protect and serve us... MOO