I don't think being a Stark(having "wolf blood")1 is the main reason for their warging abilities, I believe it's because the Starks carry the blood of the first men in their veins.
If it was just being a Stark("wolf blood")1 that gave them the abilities there wouldn't be any other wargs outside of the Starks(and and Stark relations maybe), butdescendants.
But there are wargsWildling Wargs2 north of the wall(wildlings, who also carry the blood of the first men) so that proves that being a Stark is not necessary to be a warg.
1. In other words, having the "wolf blood"
2. Wildlings also carry the blood of the first men