Firstly, that paper was written in January and examined a Stable Diffusion model that was already obsolete due to its poor training even back then. Secondly, even with that poor model they had to move heaven and earth to find a handful of examples out of hundreds of millions of training examples where they could get a blurry replica out.
Secondly, as mentioned, that model is long obsolete due to issues exactly like this. Modern models work better in part because they have better curated training sets that eliminate this sort of "overfitting." There's no indication in this article that the website in question is using one of those old models, it's just presented as a hypothetical concern.