Why isn't f(G) uniform in ECDSA?
Hasse's bound tells that the order $n$ of [that is, number of elements in] an Elliptic Curve group is about the same as the order $p$ of the underlying field [note: $x$ and $y$ coordinates are elements of the field, points $(x,y)$ are elements of the group].
For every $x$ in the field with $(x,y)$ in the group, there is exactly one other point in the group with the same $x$: $(x,-y)$.
Therefore, about one value of $x$ in the field out of two is such that there exist $y$ in the field with $(x,y)$ [and $(x,-y)$] in the group. Like, if you have $n$ nuts and $p$ persons, with $n\approx p$, and you give two nuts to each person, then only about half the persons get a nut.
Therefore, $f$ as defined in the question is not uniform in the field, for it only reaches about half of the field elements.