Can you reconstruct a simplicial set from an $\infty$-groupoid? In some categories of things with interesting structure, said structure can be recovered from the category.
For example, in the category of chain complexes of abelian groups, if you're given a chain complex $A$, you can recover its structure as a chain complex from all of the hom-groups $\hom(C_n, A)$, where the $C_n$ are all of the shifts of the complex $\mathbf{Z} \to \mathbf{Z}$.
By the yoneda lemma, the same goes for simplicial sets; you can recover the simplicial set structure of $S$ from the hom-sets $\hom(\mathbf{y}\Delta^n, S)$ and the maps between them.

I've been curious for some time if this sort of thing would help me get a handle on $\infty$-categories.
If I try the simplicial set recipe on $\infty \mathrm{Grpd}$, though, I get a problem; if I've not made an error, all of the basic simplicies are homotopic, and all of the maps homotopy equivalences! e.g. an output of this recipe could well be the constant simplicial object.
It's reassuring, I suppose, that I get a simplicial object that obviously corresponds to the object I started with, but doesn't actually help with what I was trying to do!
I suppose the essential difference between this and the previous examples is here I'm (attempting to) work in an $\infty$ category, but the previous examples were all ordinary categories. But that leaves me at a loss; I want to see how we could use the $\infty$ category structure to produce things like simplicial sets from the objects, but I'm at a loss at how to proceed.
Thus, the question: can you recover a simplicial set from an $\infty$ groupoid? And how?
 A: You can think of simplicial sets as presentations of $\infty$-groupoids; for example, in any simplicial set modeling $BG$ for $G$ a discrete group, say with one 0-simplex, the 1-simplices give generators of $G$ and the 2-simplices give relations between them. An object can have many presentations in general, and so it's unclear how to pick between them.
You might object that in the case of $BG$ there is in some sense a canonical presentation, namely the monadic one: every element of $G$ as a generator, every relation between them as a relation, etc. If you try to run the same game on an arbitrary pointed connected $\infty$-groupoid $X$ what you will write down is a simplicial space (not set), more or less the usual bar construction but using $\Omega X$ in place of $G$. 
Anyway, continuing the theme of your examples, the model-independent question would've been to ask how to recover an $\infty$-groupoid from an object in the $\infty$-category of $\infty$-groupoids, and the answer is very simple: it's the entire hom $\infty$-groupoid $\text{Hom}(1, X)$ (note that whenever you talked about a hom you implicitly only talked about its $\pi_0$). 
