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It's worth observing that "goodness," in your sense, lacks a compactness property satisfied by "adequacy" (in the sense of the linked question): a set $R\subset \mathbb Z$ is adequate if and only if for all $\epsilon>0$ and all $K\in \mathbb N$, there is a finite $R'\subset R$ such that for all $k\in \mathbb Z$ with $|k|< K$, there is an $r\in R'$ such that $|\{(r+k)x\}|<\epsilon$. In contrast, $2\mathbb Z+1$ is good, but no finite subset $R'\subset 2\mathbb Z+1$ approximates goodness: there is always an irrational $x$ such that $|\{rx\}|>1/4$ for all $r\in R'$ (take $x$ close to $1/2$).
I seem to recall that the challenge in the graph theoretic approach is simply finding a natural graph theoretic statement that implies the existence of 4-term APs in dense sets. As alluded to in your post, there are natural Fourier analytic statements that imply the existence of 4-term APs but Gowers's examples show that their proofs require a larger family of "structured" functions than is necessary for Roth's theorem. But I may be misremembering, or misinterpreting your question.
