Average first-passage times for character sums

Let $\varepsilon>0$ and, for an odd prime $p$, set $$S_\ell(p):=\sum_{n\le \ell}\left(\frac{n}{p}\right).$$ Define the first-passage time $$f_\varepsilon(p):=\min\{\ell\ge 1:\ S_\ell(p)<\varepsilon\ell\}.$$ We prove that there exists a constant $c_\varepsilon>0$ such that, as $x\to\infty$, $$\sum_{p\le x} f_\varepsilon(p)\sim c_\varepsilon \frac{x}{\log x}.$$