This is my personal opinion and partly based on the pamphlet for parents of missing children put out be the Dept. of Justice. Usually LE will put the parents out to the media for two reason IMO. Firstly, to humanize the child in case he/she is being held, secondly to observe the parents behavior for obvious reasons.
I don't have a real theory here, but I do find it interesting that Mom's letter was hand-written & I think that was to humanize Abby on how much she is cared about. Everything in this case seems very timed & calculated. For instance when her (I call) step-father was interviewed, her father, not sure on the exact dates but it seemed to rhyme to me. If you remember the case in Canada (Keenan) guessing two years ago now, police quickly figure how he was indeed abducted and had the parents talk about how much this little boy was loved by them and his brother & sisters, etc. The abductor returned the child (supposedly) unharmed after several days into his home that was under surveillance, although the family was not staying there. Very bizarre case, but it worked with a truly happy ending! IIRC the perp. was working on a construction crew in the neighborhood & thought the family was too big and wouldn't miss him or something similar.
A good point was made that LE doesn't appear to be doing ground searches. Yet, they have tossed the word "abduction" out in difference ways, since receiving the letter was announce. Personally, this is the first case (outside of Keenan) where I believe it is possible that LE really believes a child (Abby) maybe held against her will, therefore halting ground searches. If so, of course it would pertain to some info. in the letter IMO. Abby dating the letter is very fishing, it would be extremely unusual for a 14-15 year old would not do this. LE and her mother feel the letter is authentic. I also think think with the time that has passed and all the investigating that has been done, this is not someone in her circle of those she that she knows, but maybe an old acquaintance she vaguely is familiar with or internet predictor. But, if it was someone on the internet type thing, there would be tracks. Bottom line for me is whatever was in that letter, LE is not writing her off as a runaway.