This is certainly true, but I have only a statistical "proof". Using the list available on the OEIS (A055507), it appears that when you perform a $\chi^2$ test, for each prime $p=3,5,7,11,13,17,19,23,29$ and $31$, the finite sequence $\{S(n): n\le10000\}$ is uniformly distributed on $\{0,1,\dots,p-1\}$. The $p$-values are always largely above $.05$. In particular there is a quite strong statistical evidence that there are no congruence classes that appear significantly more often than others. 
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