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 $2^{n-1}.$