= Solution
For large $\sigma$, neighboring peaks overlap and the density has a broad, smooth, symmetric central hump, approximately $(\sqrt{2\pi}\sigma)^{-1}e^{-z^2/(2\sigma^2)}$ on the scale $z=O(\sigma)$. A precise central-profile statement follows from
$$
\sigma g(\sigma t)=\mathbb E\left[\frac{e^{-(t-X/\sigma)^2/2}}{\sqrt{2\pi}}\right]\longrightarrow\frac{e^{-t^2/2}}{\sqrt{2\pi}}:
$$
for each fixed $t$, $X/\sigma\to0$ almost surely and the integrand is bounded, so <dominated convergence> applies. The right panel of the preceding figure shows this Gaussian-looking core.
This approximation does not remove the <heavy-tailed distribution>. Here $\mathbb E|X|=(6/\pi^2)\sum_{n\geq1}1/n=\infty$, so the symmetric distribution has \b[no ordinary mean and no finite variance]. Adding the independent Gaussian, which has finite absolute moments, leaves these conclusions unchanged. In particular a central Gaussian sketch cannot justify assigning variance $\sigma^2$ to $Z$.
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