The set $S$ gives rise to a subscheme (which let's also denote by $S$) of $\mathbb{P}^1_{\mathbb{Z}},$ because relatively a prime pair $(x,y)$ corresponds to a section of $\mathbb{P}^1_{\mathbb{Z}}\rightarrow\operatorname{Spec}\mathbb{Z}$, and we take the union of these divisors in $\mathbb{P}^1_\mathbb{Z}$.

Now, we have a map $\mathcal{O}(d)_{\mathbb{P}^1}\rightarrow\mathcal{O}(d)_S,$ and it will suffice to show that for some $d$ some element in the image of the induced map on global sections is nowhere zero. (If one composes this map with the map $\mathcal{O}(d)_S\rightarrow\mathcal{O}(d)_{\operatorname{Spec}\mathbb{Z}}$ corresponding to a section $(x,y)$, one gets at the level of global sections the evaluation map $\mathbb{Z}[x,y]_d\rightarrow\mathbb{Z}$ on degree $d$ homogeneous polynomials.)

For this, it suffices that one can find $d$ so that this map is surjective on global sections and so that $\mathcal{O}(d)_S$ is trivial. The map will be surjective on global sections for large enough $d$ by ampleness of $\mathcal{O}(1)$, and the triviality of a power of $\mathcal{O}(1)_S$ follows from the finiteness of $\operatorname{Pic}(S)$.

It's maybe slightly inaccurate to say just that it's a case in Lemma 7.3 in the above paper; rather, this exact argument is ran in the proof of Corollary 1.3 (which immediately follows the aforementioned lemma) to prove a much more general result.