I think that's actually because she isn't really capable of envisioning loyalty to Rand, and could not comprehend the idea of someone, especially her friends, putting him before her. We will see the same thing when she expects support for her opposition to him at Merrilor.
I suspect Jordan's notes just had stuff like "Egwene impresses sisters with her take on Rand and criticisms of Elaida" and the ensuing dialogue came too much from the head of Sanderson, leading to your last point:

Indeed, one of the things I am keeping in mind as an issue coming up is how to approach those idiotic socratic dialogues, where she does not come off as clever, so much as someone in the hands of idiots. Especially with the Whites, who are supposed to be intellectually superior and whose methods of debate should have left her unable to respond (compare, for instance, their discussion with her, and the argument going on among the Whites guarding Leane in KoD, and then ask yourselves if women of the same mindset as those jailers would really have been impressed with Egwene's logic. Somehow I doubt it.