But is it really arrogance? There's no proof that Rand knows what to do in any situation - he just happens to get certain things right, and has Cadsuane to agree or disagree with him.

Egwene's reasoning is that they come from the same village, and therefore their start in life has been the same, as well as their backgrounds. Egwene has always thought she knows better than him, and he feels the same - you can see it in their fights in EoTW. Egwene thinks she knows what's best for Rand because she's always considered herself to be more mature - she was Nynaeve's apprentice while Rand was playing practical jokes along with Mat. It's just the way they grew up - Rand still acts childishly around Egwene and resists her advice more than anyone else's, and she struggles to agree with him even if he's right. The fact that she can Dream makes it worse because she thinks she knows what's best for him based on her prophecies - so she's as certain of some of his destiny as she is of the Tower's.

I still think she's there to oppose and help Rand. She's saidar personified, corny as that sounds. She'll end up being Latra Posae Decume to Rand, which will be disastrous and good.