Lemma 2.2.11 in [mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf][1] shows that the surjection you denote $sd(X) \to S(X)$ is an isomorphism if and only if $X$ is a non-singular simplicial set, i.e., if and only if each non-degenerate simplex in $X$ is embedded.  It may be interesting to determine for which $X$ the initial non-singular simplicial set under $sd(X)$ is precisely $S(X)$.

  [1]: https://mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf