This question is prompted by this one (and some of the comments that it drew).

Simplicial complex is to ordered simplicial complex as $X$ is to simplicial set. The question is about $X$.

Let $\text{Unord}$ be the full subcategory of $\text{Set}$ whose objects are the sets $[n]=\lbrace 0,\dots ,n\rbrace$ for $n\ge 0$. An object of $\text{Set}^{\text{Unord}^{\text{op}}}$ is basically a simplicial set $X_\bullet$ with a suitable $\Sigma_{n+1}$-action on $X_n$ for all $n$. The obvious functor $\Delta:\text{Unord} \to \text{Top}$ (which creates a simplex with given vertex set) determines a functor $$ \text{Set}^{\text{Unord}^{\text{op}}} \to \text{Top} $$

This has a left adjoint analogous to the realization of simplicial sets.

Is the counit of this adjunction a weak homotopy equivalence for all spaces?

If so, is this adjunction a Quillen equivalence for some model structure on $\text{Set}^{\text{Unord}^{\text{op}}}$?

If not, is there something else along these lines that works?

(This must be known. It seems like an obvious question, and from some comments at the other question I gather that at least on the homology side this is something people thought about a long time ago.)