Do the noble families in Westeros have to be so absurdly long-lived for the story to work? Couldn’t “Stark” just be a name adopted by whoever’s ruling the North at the moment, and Ned’s actual family only go back a couple of hundred years?

For most of them, no. Although note that most of the Great Houses, to take an example for ease of discussion, are of more recent vintage: the Martells as they exist today date back to Nymeria’s Conquest, the Tyrells only became prominent under Mern VI toward the end of the Gardener dynasty, the Tullys only became prominent with Aegon’s Conquest, the Baratheons are only 300 years old, the Arryns date back only to the Andal invasion, and the Lannisters are Andal transplants who married into the name when the original First Men line ended with Gerold III. And the Greyjoys claim to have ancient kings in their blood, but only recently began ruling the Iron Islands. 

But for the Starks, yes. Because they have to be of the blood of Bran the Builder, to be wargs and greenseers descended from the daughters of the Warg King and the Marsh King, because of what Winterfell is for.