A normal p-subgroup acts trivially on every simple module in characteristic p

Restrict $S$ from $G$ to the normal subgroup $N$. By [L2], the restricted module has finite length, so it contains a minimal nonzero $N$-submodule $T$. Then $T$ is simple as an $N$-module by [F1].

Since $N$ is a finite $p$-group, [L1] makes $kN$ local. Its augmentation quotient is the trivial simple module $k$, and every simple module over a local finite-dimensional algebra is its unique simple quotient. Hence $T$ is the trivial $kN$-module. Therefore the fixed-point space $S^N:=\{s\in S:ns=s\text{ for every }n\in N\}$ contains $T$ and is nonzero. Because $N$ is normal in $G$, the subspace $S^N$ is $G$-stable: for $g\in G$, $n\in N$, and $s\in S^N$, one has $n(gs)=g(g^{-1}ng)s=gs$.

The nonzero $G$-stable submodule $S^N$ must equal $S$ by simplicity of $S$. Hence every element of $N$ acts trivially on every vector of $S$.