Cz:
I don't think LE, WS sleuthers, Find Hailey FM members, truckers on HW 120, or NG viewers have stopped searching for Hailey as a result of frustration with inconsistencies and recollections of Billie and/or Shawn. In my opinion, if Billie were to hold a press conference to set the record straight and lay out the whole story, it would come across as her focusing on her image again, rather than Hailey. Also, it could look like she is trying to get the public on her side vs. LE's side. That would be a bad perception to perpetuate. It has to be a team effort, as I'd guess Marc Klaas advised her.
I personally believe that Billie should do everything you suggested with LE/FBI behind closed doors. Tell her whole story, the whole truth and nothing but, no matter who looks good or bad, because Hailey is more important to her than what anyone thinks. The public hasn't stopped searching because of Billie, those of us that are trying to fit all of the pieces together just get roadblocked, imo, by her inconsistent statements. It makes it hard for us to move forward without the basic foundation of the situation and a time line. We still don't know when she was exactly last seen. From statements made recently, LE appears to be struggling with these same challenges.
If Billie were to proactively suggest an all day session with LE to tell her whole story and answer any and all questions, it would go a long way towards her credibility, imo. Taking the poly would also help.
I understand that you and she don't want people to think bad of her (though I disagree that that's what's blocking efforts to find Hailey). On that front, she could ask that LE tape all of the session so it's on record and she can make it known that she has fully cooperated and will continue to do so, in good faith. She could then circulate a press release to all media outlets confirming that she wants to work as a team with a LE so that Hailey can be found as quickly as possible; noting that she asked for and voluntarily met with them to provide any additional information required to help move the investigation forward. The press release should also include a plea to the public to continue search efforts and keeping an eye out for Hailey.
This approach would truly help move the investigation forward and would improve her credibility with the public, in my opinion.