Why do you think Sansa was so under prepared for life in the capital? For all her love of songs and courtesies, she has zero political education. Cat was her father's heir until her brother was born and ran Riverrun for years. Why was Sansa's education so lacking when compared to Robb or even Bran?
Actually I tend to think that there was a problem in the political education of all the Starklings, including Robb, and that this was informed by Ned’s desire to just be left with his family in the North. And while age is a perfectly good explanation for that in the case of the younger siblings, it’s utterly bizarre with Robb, being that he was only two years shy of his majority when the novels start, and that his status as heir and the future Warden of the North necessitates a deeper understanding of politics.
Sansa’s story serves to highlight the problem more than any of her siblings’
because her story is the most political, and because she is the one Starkling who was thrust into
the heart of politics in royal court with absolutely no backup and no
one to guide her. Robb had Catelyn, the Blackfish and his experienced bannermen around to advise
him; Bran had ser Rodrik and Maester Luwin; even Jon had Jeor Mormont
and Donal Noye and Maester Aemon; who did Sansa have? Septa Mordane had
no experience and no understanding of politics and Ned was waist-deep in
trying to pick up Robert’s slack and figure out the truth about Jon
Arryn’s death, and it’s not like he was the best political player anyway
as demonstrated by his failure to utilize the power of his office. Which left Sansa on her own to act within her limited
experience, and was exacerbated by her idealistic views, and her beauty
= integrity and worthiness perspective.
But Sansa’s isn’t simply a problem of education - that I can at least explain by her age and the fact that her education was no where near done by the time she came to King’s Landing; no, the problem is that Sansa was left to her own devices in the capital with no effective mentor and no direction or even basic explanation of their situation. Her betrothal meant she would be in direct and constant contact with both Cersei and Joffrey, something she was utterly unprepared for and yet she was left to figure things on her own. The situation with Lady, Ned’s guilt over it and the resulting distance from Sansa, and Sansa’s own need to rationalize it that led to her suppressing her own memories only made her even more vulnerable.
Now I wouldn’t say that Sansa had zero political education; as a girl, Sansa’s
political education was vastly different from anything her brothers’
received and centered on completely different things, so her political
action took shape in a different way. Things like courtesies, proper conduct and
attire, and a lady’s role in a household are part of a
girl’s political education since they are often the only tools afforded
to women in Westeros, and indeed we see Sansa utilize them effortlessly in the aftermath of Ned’s arrest as an intuitive employment of her education in carefully constructing her words and image. She was taught the importance of one’s image and cultivating it in a way that inspires loyalty, but she was not taught that a political image can be only that- an image. Which meant that she considered things like proper etiquette and beauty and highbirth to be signs of a good and worthy person, an idealistic (and elitist) view that was encouraged and rewarded by her Septa, and wasn’t challenged by Catelyn.