They are whiter than white. Ned refuses to dislike the people who murdered his father ac 2 Barristan "he never meant your family any ill"; borderline unnatural behavior BTW. He indulges and encourages his daughter Arya. She herself is portrayed as being willful and stepping outside traditional female roles to make her own identity. You then have the Sansa, the perfect disney princess, period. So much so in fact that her simply singing and showing her innocent purity manages to shame the hound and causes him to help Arya in the end. Meanwhile, Rand al...sorry, Jon Snow, is shown as being without prejudice. He accepts wildling culture as superior, he lets women fight, he is fine with giving a gay man responsibility and he is opposed by incompetent bigots in his story. Catelyn lives by the motto "family, duty, honor". Rob Stark wants to free his people from oppression and there's nothing like a bit of romantic nationalism to get readers sympathy. He even gets to fight the people who murdered his family.
For me, grey means I cannot tell who to root for. It does not mean "good people can lose", which I imagine a lot of people would reply to this with. The Starks are good but this doesn't work in the real world. I don't really see how that matters. Whether something is grey is about morality and that point just doesn't seem to be a matter of morality. Its just that bad things can happen to good people. There is no attempt to subvert the morals of the Starks or create ambiguity for the readers.
This is to such an extent that we actually know Tyrion is a good guy because he is sympathetic towards the Starks (even though they were as prejudiced against him for being a dwarf as everyone else). Lord forbid that Tyrion would dislike the family who abducted him because they assume he is a scheming dwarf. We also have Daenerys identification of the Starks as enemies being demonized by morally upright characters like Barristan Selmy. Lord forbid that Dany would dislike the Usurpers principle military backer. A character who criticizes the moral purity of the Starks is almost always shown to be bigoted and incorrect.