The lower central series of $S_4$ is given by : $$\gamma_1 =S_4\ge \gamma_2=A_4\ge\gamma_3=A_4\ge \gamma_4 =A_4\ge... $$
This series is clearly central as each $\gamma_i/\gamma_{i+1}$ is central in $S_4/\gamma_{i+1}$ but $S_4$ is known to be not nilpotent. While we know that a lower central series of a group $G$ is central if and only if the group $G$ is nilpotent, which means that the series starts at $G$ and terminates at $\{1\}$ at some finite step. Is there anything I'm missing in this reasoning. Thanks for your help!
A finite group $G$ is nilpotent if and only if the lower central series ends at $1$, if and only if the upper central series ands at $G$. Since $S_4$ has trivial center, it cannot be nilpotent. Consequently, its lower central series cannot and at $1$ - and it doesn't, as you have shown.