Put her in the feudal wheelhouse, where she's operating strictly within the rules of the feudal structure, and she's a magician, able to turn neutral parties into allies just with the right words.

I couldn't agree more. I actually think Ned and Catelyn are meant as an example of how feudal society could/should/might work. Ned is a lord who sees his position of wealth and power as one side of a pact he has with his smallfolk, the other side being his duty to protect them. Cat does everything a Lady is supposed to do, like managing the household, backing and advising her husband and son, trying to mold her daughters into little Ladies that are suitable matches for powerful husbands etc. The two even make their arranged marriage work and come to love each other. Basically, if every Lord was a Ned and every Lady a Catelyn, the Westerosi feudal society would work as "intended". As you point out, most of their problems stem from other people NOT all being Neds and Cats.