It seems that the paper The Maximum Multiplicity of a Generator in a Reduced Word is the state of the art concerning this problem:

They write $\mathcal{M}(k,n)$ for your $S(n,k)$ and show that for fixed $k$ and $n \rightarrow \infty$ one has $\mathcal{M}(k,n) = c_k n + p_k(n)$ for a constant $c_k$ and a periodic function $p_k$. In particular, it grows linearly in $n$.

(Disclaimer: I have not read the paper, but only reproduce their abstract here.)

It seems that the paper The Maximum Multiplicity of a Generator in a Reduced Word by Christian Gaetz, Yibo Gao, Pakawut Jiradilok, Gleb Nenashev and Alexander Postnikov is the state of the art concerning this problem:

They write $\mathcal{M}(k,n)$ for your $S(n,k)$ and show that for fixed $k$ and $n \rightarrow \infty$ one has $\mathcal{M}(k,n) = c_k n + p_k(n)$ for a constant $c_k$ and a periodic function $p_k$. In particular, it grows linearly in $n$ for $k$ fixed.

They do not seem to give an anwer for the maximum over all $k$ though.

(Disclaimer: I have not read the paper, but only reproduce their abstract here.)