The game of thrones has always been a treacherous game of great power and privilege — one only noblemen dare to play. However, the great houses that have ruled over Westeros for thousands of years are growing increasingly thin on Game of Thrones.
For what it’s worth, the Baratheon line is now extinct. Tommen Baratheon, First of His Name, may sit on the Iron Throne, but we know he’s not King Robert’s legitimate heir. (That, and it won’t be long until Cersei’s last son meets his prophetic end.) In fact, the only Baratheon heir to make it out of the game of thrones alive is Gendry, but he’s an illegitimate bastard. Similarly, the Martells are pretty much dead following the assassination of both Doran and his son Trystane by Oberyn’s bastard daughters. The Tyrells are also in bad shape, with Mace Tyrell’s only son (on the show) being Loras — and it will be a miracle if Loras makes it out of Season 6 alive. Following the death of Ramsay Bolton, the Boltons have been eradicated from the realm. Meanwhile, other once-great houses, like the Tullys, are no longer even houses.
And in the most recent episode of Game of Thrones, titled “Battle of the Bastards,” we can now add the Starks to the growing list of endangered houses. When Ramsay Bolton shot an arrow through Rickon Stark’s heart, he may have sealed House Stark’s fate. There are only three legitimate Starks left to carry on the name — and Sansa and Arya are both women. Jon Snow, however, would need to be legitimized by a king if he ever wanted to become a Stark.
Then there's Bran, whose condition complicates things. In George R.R. Martin’s A Game of Thrones, following the accident that paralyzes Bran from the waist down, Ned Stark laments that his young son will never be able to have children. Now, that could just be a father's grief speaking, but since Maester Luwin never said anything officially, there’s no way of really knowing if everything’s A-OK down there. (It’s doubtful even Bran knows, unless he had time to ~ experiment ~ in the Three-Eyed Raven’s cave.)
So, if Bran can’t produce an heir and Jon Snow is nothing more than a bastard, then who will carry on the Stark name? Ned is dead. His brother, Brandon, is dead. And his other brother, Benjen, is an undead, and as far as we know, they can’t produce the way warm bodies do. That, of course, leaves Sansa.
While it’s not commonplace for women in Westeros to carry a noble line — except for Dorne, where women are actually celebrated leaders — it can be done. The freefolk legend of Bran the Daughterless comes to mind. In the books, Ygritte tells the tale of Brandon Stark, supposedly a former Lord of Winterfell, who could produce no heir save for a daughter, and at this time (the exact time is never specified, but we know it was "a long time back"), the Starks were on the verge of extinction. However, a great freefolk raider named Bael the Bard disguised himself as a singer and seduced and impregnated Brandon's only daughter. She gave birth to a bastard son who eventually inherited Winterfell.
If the tale of Bran the Daughterless is true — though he's never officially mentioned in the old chronicles of Winterfell — that means Sansa can sire a new generation of Starks. After all, she probably wouldn't be the first noblewoman to do so. Lyanna Mormont of Bear Island and Daenerys Targaryen are both in the same boat. Of course, she'll probably have to marry a man of lower birth. (Like, say ... Littlefinger. After all, his surname is Baelish.)
However, the Starks have even bigger problems than continuing their family lineage. The remaining Starks no longer control any of their lands, castles, or wealth, and their forces were all but destroyed by the Boltons. They're pretty much powerless at this point, and as Jon warns Sansa in the promo for the Season 6 finale, they have a lot of enemies.
Not to mention, by the time Daenerys takes Westeros and abolishes its age-old class system, Sansa might not even need to worry about giving birth to an heir. By then, even women in the realm will rule -- and they'll have the right to choose what to do with their own bodies. That is what you meant by making Westeros better than how you found it, right Dany?