I think something equivalent to this has been studied in the combinatorics literature.

A CW complex of course has a poset of faces. If the CW complex is regular, then the order complex of the face poset is homeomorphic to the complex. So (assuming regularity, which I haven't checked), your question is equivalent to ordering permutations by subword inclusion up to deletion and monotone reordering. A useful keyword for such studies is permutation patterns.

The lattice of permutations ordered by pattern containment has been studied by Jason Smith. See, for example, the paper

Smith, Jason P., A formula for the Möbius function of the permutation poset based on a topological decomposition, Adv. Appl. Math. 91, 98-114 (2017). ZBL1370.05227.

I think something equivalent (or at least closely related) to this has been studied in the combinatorics literature.

A CW complex of course has a poset of faces. In this case, this poset is obtained by ordering permutations by subword inclusion up to deletion and monotone reordering. The keyword used in the combinatorics literature for this sort of subword inclusion is permutation patterns.

Now, if a CW complex is regular, then the order complex of the face poset is homeomorphic to the complex. As you point out in the comments, the simplicial set has $n!$ faces of dimension $n$, and in particular has a single vertex. So the simplicial set isn't regular, as I'd initially thought it might be, and your question doesn't reduce directly to this poset. It certainly seems like the two objects should be closely related, however.

In any case, the lattice of permutations ordered by pattern containment has been studied by Jason Smith. See, for example, the paper

Smith, Jason P., A formula for the Möbius function of the permutation poset based on a topological decomposition, Adv. Appl. Math. 91, 98-114 (2017). ZBL1370.05227.

That paper cites also his earlier papers on the topic.