The arguido interview had nothing at all to do with finding Madeleine. The arguido interview was to get something to enable the police to lay charges against her. Anyone who would defy legal advice to answer under those circumstances is plain stupid ... Gerry was lucky to get away with it ... but I have always been of the opinion that they weren't interested in Gerry anyway ... Kate was the target.
I can sort of agree with your last comment. They zeroed in on her. But why? Let's remember, at this point it is in everyone's interest to find Maddie alive and well. It would be the result of the century for everyone. There is no motive at this point of proceedings for PJ to embark on some sort of vendetta - it's a missing little girl.
Inept / incompetent / buffoons or not, they clearly weren't buying what she was selling. And when she refused to answers questions, her lawyer placed the tin hat on that perception.