After the cops discover the victims body, the captain states that Elsbeth got her wish and they will be searching both Margo and Lainey's houses. You would think that's the progress she wanted. Secons later however Elsbeth automatically assumes they will focus only on Lainey and not on Margo. Why?

What evidence does she have that the cops will completely ignore a possible suspect? Which happens to be the last person to see the person alive. Which happens to have a missing rug large enough to dispose of a body. Which happens to order a new rug the day after the victim went missing.

Later the cop says "I'd love to make that woman a suspect...but I can't do it" - why? What reasons does he have? Even though they found rug fibres in the fire place (how the f do you burn such a large rug in such a tiny fire place?!). Even though she has a history of impulse violence as can probably be concluded by her criminal record.
And of course the only evidence pointing towards the wife is found by Elsbeth...ugh. And they even state that this note is only circumstantial.
So the cop ignores the evidence he found himself and considers the circumstantial evidence Elbseth found the only possible lead...

So they find out that Margo and the victim where in the same jail cell just before his deception started...nope not enough evidence.
They find out that a brass statue is missing at Margos house, and they know that the victim was killed with something made of brass...nope not enough evidence.

So yeah, the cops obviously don't think of Margo as a suspect until Elsbeth solves the case...even the worst cop in the world wouldn't be so stupid.