It’s not hard to check that $\Omega$ is trivially cofibrant (and so its homotopy type is trivial). Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the property that any subobject $A' \to A$ extends to a subobject $B' \to B$ such that $i^*B' = A'$; when $i$ is mono, this is always possible, with canonical solutions given by the direct image $\exists_i A'$ and the dual image $\forall_i A'$.
Nothing here is special to simplicial sets: this argument anpplies equally in any model structure on a topos in which all cofibrations are monomorphisms.