Non-zero winding number on a space curve implies a linked curve in the zero set? The following statement has been largely improved from my original post thanks to discussions with @DmitriPanov ad well as the comment from @Wojowu.

Let $f \colon \mathbb{S}^3 \to \mathbb{R}^2$ be real analytic.  Let $C$ be a closed space curve in $\mathbb{S}^3$, which I might need to assume to be unknot.
   If $f(C) \subset \mathbb{R}^2$ has a non-zero winding number around $0$, then $f^{-1}(0)$ contains a space curve in $\mathbb{S}^3$ that is linked to $C$.

I consider this as a generalization of Kronecker's existence theorem.  I sketched a plausible argument in my comment to the answer of @DmitriPanov, which I did not check with great care.  The statement and argument is obviously generalizable to other dimensions, using topological degree.
I believe that this result is not new.  So where do I find a reference for this and its higher dimensional versions?
 A: Now the statement is indeed correct. What follows below is and answer to the previous version of the question which I'll keep for the moment.

Note that the condition for $f(C)$ to have non-zero winding number around $0$ is more-less empty. Indeed, to fix the winding number of $f(C)$ around $0$ one needs to choose infinity. And for more-less any closed curve $\eta$  in $S^2$ that doesn't pass through $0$ one can choose $\infty \in S^2$ so that the winding number of $\eta$ in $S^2\setminus \infty$ around $0$ is non-zero. 
Now, for a concrete set of counter-examples suppose that $f: S^3\to S^2$ is ANY map with non-zero differential at a point $x\in S^3$ such that $f(x)\ne 0$. Then take a small ball $U$ containing $x$ such that $0\notin f(U)$ and a curve $\gamma\subset U$ that projects to a small circle in $S^2$. Choose $\infty\in S^2$ such that $f(\gamma)$ separates $0$ from $\infty$. Then clearly the winding number of $f(\gamma)$ around $0$ is $\pm 1$ but $f^{-1}(0)$ doesn't contain a component linked to $c$.
I can see only one way to fix this. Ask $f$ not to be null-homotopic and ask $C$ to be a full premiage of a point $x\in S^2$ different from $0$ ...
