I'm flummoxed too! Either this crime was meticulously planned to make it virtually impossible to find her, or, there is a piece of the puzzle we just don't know/are not being told.
I tend to think its the latter.
LE must strongly believe that the perp(s) are local and that Holly is still in the area, safe or not. Otherwise, we would have seen more active involvement of the FBI. I have not heard of any nationwide pleas to BOLO for Holly.
Also, there have been no media reports (or rumors) I have seen or read that even suggest any true LE searches via search warrant. In TN, grand jury proceedings, including search warrants, are secret until an indictment comes down and the perp is picked up. However, given the heightened number of media on the ground, someone would likely see a SWAT team type LE presence if any "hot" property was searched.
On the other hand, if they have ANY idea of who it could be, and if they have ANY idea that Holly would be wherever they think he might be, it becomes a rescue, and no search warrant is needed. Locally, we had a brutal torture-murder of a young couple, and had LE and the family fit the clues together a bit sooner, the girl might have been found alive, so I'm sure LE is plenty aware that any suspicion of where Holly might be must be acted on asap.
If they have the perp(s) DNA, they know by now that he is not in the national system or else there would have been warrants for his arrest out with nationwide publicity. But they appear to be placing almost all their marbles on the Parsons' area.
I think LE has convinced the family and about everybody else locally that "loose lips sink ship." Especially if it is someone local, they are going to have to find him before he finds out they are after him. Or at least they want it that way.
And as you put it, its a puzzle. LE hasn't solved it. We don't have near the pieces that LE does. And it looks like we won't know anything until they want us to. Or its leaked.