I’d like to point out some comments out of the books about Sansa and her ability to get involved with the politics of Westeros.
She is good at this, he thought, as he watched her… She would have made Joffrey a good queen and a better wife if he’d had the sense to love her.
Oh, look, a fan favorite character clearly stating that Sansa is good at being friendly and diplomatic to other political people (this was at the PW). He also notices her potential, what she could have become if her family and future hadn’t fallen apart.
"Sometimes the best way to baffle them is to make moves that have no purpose, or even seem to work against you. Remember that, Sansa, when you come to play the game."
"What… what game?"
"The only game. The game of thrones."
Here we have another master schemer and generally appreciated character talking to Sansa about politics. He also seems to take it for granted that she will come into her own and be a part of the political realm of Westeros at some point. He’s even giving her advice, training her for this future role.
If fans claim to love Tyrion or Littlefinger but mindlessly bash Sansa, maybe they should listen to their favorite characters more closely. She just might be the one to take them down in the end.
I agree with your main argument but I can think of other examples which call out more strongly as to Sansa’s potential in interacting with the players. Yes, she was good at interacting with the lords and ladies in King’s Landing. She showed her kindness and compassion towards them and the things going on in their life but she was not acting in a political way..To me what Tyrion observed was an extension of Sansa’s view on how court life works. It made me think of that quote:
If I am ever a queen, I’ll make them love me.
There was no agenda for Sansa in those interactions. Likewise, I am skeptical as to how much helpful advice Littlefinger was truly offering. To me, those first conversations between Sansa and Littlefinger were his way of showing how clever he was. The real lessons were learned later on when she reached the Eyrie and particularly after Lysa’s death.
She figures out how Littlefinger gained Lord Nestor’s support and how the way in which he did it ensured his continued support. It is at that point she realises the lesson Lies and Arbor gold and not only learns it but puts it into play.
"I am Alayne, Father. Who else would I be?"
She told Littlefinger what he wanted to hear. She kisses him when expected to, she manages to play the role of a highborn bastard girl. The tools she is learning now will serve her in the politics of Westeros. The best players know how to put on a different face and now so does Sansa. It will be interesting to see how she applies her new skills once she no longer has to pose as Alayne Stone.