Except that the Aes Sedai are revered and honored in the borderlands. And many in Tar Valon (largest city in the Westlands)think they are pretty OK too. So not all are "incredibly suspicious of them". If you were going to be accurate you'd probably have to say "In some lands most people hate ...." I'd say that in Andor, most people are "somewhat" suspicious of AS not "incredibly suspicious". Same can be said of Cairhien. Note that both Corgaide Marendevin (Keeper of the Keys at the Cairhien Palace) and Renee Harfor (same for the Camelyn palace) show remarkable deference and genuine respect to the AS for someone who is "incredibly suspicious" of them. Contrast Renee Harfor's demeanor towards AS vs her demeanor towards the mercinary captains for example.
As to why anyone would want an AS advisor, you have to ask many of the Kings and Queens of the Westlands. All of the Borderlander rulers have AS advisors, the Queen of Andor had one, so did many of the rulers of Cairhien. Tylin Quintara the queen of Altara had one until she was recalled to the WT after Elaida's coup, and Tylin liked her. As does the Dragon Reborn his own self. Don't know of any rulers who have a Whitecloak Advisor.

NaCl(if I was going to be "incredibly suspicious" of someone it would be the Whitecloaks)H2O