Give each element an integer weight with the entire set having weight $2k+1.$ Take all sets with total weight at least $k+1.$ Your case was one weight equal to $1$ and the rest $0$.
Take the $7$ lines of a Fano Plane and anything containing one of them. That can't be achieved by a weight function.
An intersecting family of smaller size can be enlarged to one of size $2^{n-1}.$
Also, given an intersecting family of size $2^{n-1}$ one can take any inclusion minimal member and replace it by its complement. This kind of switching can take you from any one to any other. The Fano plane example arises in this way for starting with all sets with $4$ or more of $7$ elements and then switching seven of the minimal sets.