They did use a sketch artist. That's how they came up with the guy Edmunds.
They persuaded the milkman(a minor hero) to pretend to have identified the person. The reason is obvious, and if you were in the same situation, it would be foremost on your mind, too.
Protect Elizabeth and Mary Catherine. He told Mary Catherine that if she told, he would hurt Elizabeth. The sketch was shown to everyone except Mary Catherine? Puh-leeez! even the least competent yokel deputy could do better than that.
The whole family polygraph thing is a ruse as well, to make the perp think that the family has witheld the truth from the police, as directed.
The note was probably more than just ransom, maybe not ransom at all. Some psychobabble about "loving" Elizabeth etc...
Hence the "wonderful story" schtick. "We understand that you love her and can see it from your point of view.." And the family in front of the cameras on and on about loving her -- a profiler's suggestion about how to get to the perp. Maybe the perp thinks the family doesn't appreciate her...
Get it? I just hope he doesn't lurk here.
Good thinking. Makes sense to me.
But you don't think Edmund is the perp, do you? I do not.