Several points here:

I agree that Rand doesn't care about Ajah politics and that her main point was to impress on AS leadership that as Amyrlin should would pay attention to Ajah functions and prerogatives

BUT I also think she was thinking logically about Rand's response to who was sent. Obviously her point with the Grey makes sense, but she has no reason to believe that Rand has had negative association with the Green... she has no clue about what's happening with Cadsuane, and she left before Kiruna, Alanna and Bera screwed the pooch royally with Rand (I don't recall if she knows about the Bonding) ... so she's going strictly on what sending a Green should mean, a woman who likes men ... hopefully she'll do a little digging before blindly sending a Green with the group.

Her knowledge of Rand is that the Blue is the "good Ajah", I agree that I would send Pevara or Tarna to Rand based on personality/capability, but realistically any male Channeler is going to be weary of a Red. And let's think about Rand's experience with Galina... Egwene may have some clue about that since the Tower was abuzz with what happened there!

I totally agree with you on your opinion about using their friends to approach him. Nynaeve, and Elayne would be the logical choices on approaching him ... or at the very least the best way to get up to date information on his current state of mind!