**Disclaimer.** This post is a partial solution to my problem.

---

>**Claim.** *Suppose $X$ has density and  $\mathbb E\,\|X\|_q \lt \infty$, where $q$ is the Hoelder / harmonic conjugate of $p$. Then, $L(S_p^n,t) \longrightarrow 0$ in the limit $t \to 0^+$.*

Indeed, if $X$ has density, then for every $w \in \mathbb R^n$, the random variable $X' w$ also has density, and hence a continuous CDF. 
Let $R$ be a large positive number. By the preceding argument function $t \mapsto \underset{w \in S_p^n,\,|u| \le R}{\sup}\mathbb P(|X'w-u| \le t)$ is continuous, and so 
$$
\tag{1}
\lim_{t \to 0^+}\sup_{w \in S_p^n}\sup_{|u| \le R}\mathbb P(|X'w-u| \le t) = \sup_{w \in S_p^n,\,|u| \le R}\lim_{t \to 0^+}\mathbb P(|X'w-u| \le t) = 0.
$$

On the other hand, if $|u| \gt R$, then $|X'w-u| \ge ||X'w| - |u|| \ge |u| - |X'w| \ge R-|X'w|$, and so
$$
\begin{split}
\sup_{w \in S_p^n}\sup_{|u| \gt R} \mathbb P(|X'w-u| \le t) &\le \sup_{w \in S_p^n,\,|u| \gt R}\mathbb P(|u| - |X'w| \le t)\\
& \le  \sup_{w \in S_p^n,\,|u| \gt R}\mathbb P(R - |X'w| \le t)\\
& \le \sup_{w \in S_p^n} \mathbb P(|X'w| \ge R - t)\\
& \le  \mathbb P(\sup_{w \in S_p^n} |X'w| \ge R - t)\\
&= \mathbb P(\|X\|_q \ge R - t)\\
&\le \frac{\mathbb E\, \|X\|_q}{R-t} \lesssim \frac{1}{R} \overset{R \to \infty}{\longrightarrow} 0.
\end{split}
\tag{2}
$$

Combining (1) and (2) completes the claim.