I have never thought of a counter example, but I would not bet on a positive answer to the first question. However, the answer to the second question is yes: this version of the singular functor is a right Quillen equivalence for a suitable model category on $\mathrm{Set}^{\mathrm{Unord}^{op}}$.

As mentioned above by Mike Shulman, this homotopy theory of symmetric simplicial sets is studied in the paper

J. Rosický and W. Tholen, Left-determined model categories and universal homotopy theories, Trans. Amer. Math. Soc. 355 (2003), 3611-3623.

In their paper, they construct the model structure on $\mathrm{Set}^{\mathrm{Unord}^{op}}$, but this has to be completed by

J. Rosický and W. Tholen, Erratum to "Left-determined model categories and universal homotopy theories", Trans. Amer. Math. Soc. 360 (2008), 6179-6179.

in which the authors explain that, after all, the class of cofibrations they considered is not the class of all monomorphisms (this is why I have some doubts about the fact that the topological realisation would be well behaved if we do not consider a cofibrant resolution in the picture).

You may find another construction of this model category (with an explicit description of what the cofibrations are) as well as its precise link with Kan's subdivision functor in section 8.3 of

D.-C. Cisinski, Les préfaisceaux comme modèles des types d'homotopie, Astérisque 308 (2006),

as an example of a test category in the sense of Grothendieck; see also this MO question.

I have never thought of a counter example, but I would not bet on a positive answer to the first question. However, the answer to the second question is yes: this version of the singular functor is a right Quillen equivalence for a suitable model category on $\mathrm{Set}^{\mathrm{Unord}^{op}}$.

As mentioned above by Mike Shulman, this homotopy theory of symmetric simplicial sets is studied in the paper

J. Rosický and W. Tholen, Left-determined model categories and universal homotopy theories, Trans. Amer. Math. Soc. 355 (2003), 3611-3623.

In their paper, they construct the model structure on $\mathrm{Set}^{\mathrm{Unord}^{op}}$, but this has to be completed by

J. Rosický and W. Tholen, Erratum to "Left-determined model categories and universal homotopy theories", Trans. Amer. Math. Soc. 360 (2008), 6179-6179.

in which the authors explain that, after all, the class of cofibrations they considered is not the class of all monomorphisms (this is why I have some doubts about the fact that the topological realisation would be well behaved if we do not consider a cofibrant resolution in the picture).

You may find another construction of this model category (with an explicit description of what the cofibrations are) as well as its precise link with Kan's subdivision functor in section 8.3 of

D.-C. Cisinski, Les préfaisceaux comme modèles des types d'homotopie, Astérisque 308 (2006),

as an example of a test category in the sense of Grothendieck; see also this MO question.