I can give a proof of stability.

As Deligne notes in the very last paragraph, in the etale setting $p_\#$ is $p_!$ up to a shift and twist: For a smooth morphism, $p^*$ and $p^!$ agree up to a shift and twist, and $p^!$ has a left adjoint $p_!$, which thus agrees with $p_\#$ up to the dual shift and twist.

Since shifting and twisting are equivalences of categories, it suffices to check that $p_! s_*$ is an equivalence of categories. Now $s_*$ of any sheaf is compactly supported over the base $X$ (since a section is compact), which means $p_! s_* = p_* s_*$, and $p_* s_*$ is the identity, and thus an equivalence, by the Leray spectral sequence.