Faille is a little naive at times, but she is behaving the way a Saldeaen woman is expected to behave. If she was unique, Elyas' advice to Perrin would not have worked. I have posted in other threads about the gift of knowledge a reader gets from seeing multiple viewpoints. However, if you were only allowed to read Perrin's point of view, there's a good chance you wouldn't figure it out any more quickly than he did. Especially if you were brought up to think of women as physically weaker and therefore in need of protection. She hides her spying from Perrin because she knows he will not be rational about it. She is much better at getting information than he is, at being a leader than he is, and he must learn to trust and respect her, which means he must treat her as an equal. You only coddle those you feel are inferior to you, so she is insulted. Just like Mat must learn to treat women as more than objects or nuisances, and just Rand must realize that a women's death is no more grievous than a man's. In essence, they must learn to treat women as equals, just like the women need to learn the same truth about men. Also remember that Perrin's rescue of Faille resulted in the destruction of the Shaido, and yet another alliance with the Seanchan. In spite of his seemingly narrow-minded goal, he succeeded in doing much more. Had he given up on Faille, the Shaido would have continued to create chaos, which only helps the Shadow.
By the way, your assumptions are proving my point. Many people just will not change their opinions no matter the evidence. Perrin and Faille are guilty of this.