Showing the non-pseudorandomness of a generator
This is an attempt to answer my own question based on Squeamish's helpful comments. If terminology is unclear or misleading, please tell me!

Let $D(s) = \mathbb{I}\{s = x||G(x), x \in \{0,1\}^n\}$. Then, if our putative PRG is $G'$, $\Pr[D(G'(s)) = 1] = 1$ - that is, $G'(s)$ is always in the format that $D$ looks for. However, $\Pr[D(r) = 1]$ (where $r$ is uniform from $\{0,1\}^{2n}$) is $\frac{2^n}{2^{2n}}$, to reflect that $s||G(s)$ has a range of only $2^n$ - being completely determined by $s$ - while there are $2^{2n}$ possible values $r$.

Thus $|\Pr[D(G'(s)) = 1] - \Pr[D(r) = 1]| = 1 - \frac{2^n}{2^{2n}}$, a non-negligible function.