This is an answer to the edited question.
First, observe that the composition of functors $\def\N{{\rm N}}\def\Sing{{\rm Sing}}\N∘\Sing$ in the main post computes
the homotopy colimit of the simplicial object $[n]↦(\Sing(G))^n$.
This can be seen as follows.
Observe that the functors $\Sing$ and the homotopy coherent nerve are Quillen equivalences.
After the composition $\N∘\Sing$ we also implicitly apply
the left Quillen functor given by the identity functor on the underlying categories from the Joyal model structure to the Kan–Quillen model structure on simplicial sets.
(Indeed, we need to compare the value of $\N∘\Sing$ to a space (like $\B G$), not an (∞,1)-category.)
Altogether the whole composition is a homotopy cocontinuous functor.
Such a functor can be uniquely specified (up to a contractible choice) by its values on generators and maps between them,
which in our case correspond to categories of the form $\{0→1→2→⋯→n\}$ for all $n≥0$.
As one can see from the definitions, in our case all these values are contractible spaces.
After we apply $\Sing$, we are free to apply a different Quillen equivalence, and then take the corresponding (unique up to a contractible choice) homotopy cocontinuous functor to simplicial sets.
For example, we can apply the right Quillen equivalence given by the usual enriched nerve functor, which goes from simplicial categories to Segal categories, and then apply the left Quillen equivalence given by the inclusion of Segal categories into complete Segal spaces, i.e., simplicial objects in simplicial sets equipped with a certain model structure.
(See Theorem 8.6 in Bergner's Three models for the homotopy theory of homotopy theories.)
The enriched nerve functor yields the simplicial object $[n]↦(\Sing(G))^n$.
The generators $\{0→1→2→⋯→n\}$ correspond to simplicial objects in simplicial sets given by representable presheaves of simplices.
The unique homotopy cocontinuous functor that sends these generators
to contractible space is simply the homotopy colimit functor.
Indeed, the homotopy colimit functor is homotopy cocontinuous
and it sends all representable presheaves to contractible spaces.
This proves the above claim.
Denote the above homotopy colimit by $\def\B{{\rm B}}\B(\Sing(G))$.
What is the easiest way to see that principal $G$-bundles over a topological group $G$ are classified by $\B(\Sing(G))$?
One way is provided by Theorem 8.5 in arXiv:2203.03120 (see also Theorem 1.1 in arXiv:1912.10544),
which gives us an explicit formula for the classifying space of any homotopy coherent sheaf $F$ of spaces on the site of topological spaces equipped with numerable open covers.
(Recall that any open cover of a CW-complex is numerable, so this is a strictly more general setting than in the original post.)
The classifying space is given by the homotopy colimit of simplicial sets
$$\def\hocolim{\mathop{\rm hocolim}}\def\op{{\rm op}}\def\gs{{\bf Δ}}\hocolim_{n∈Δ^\op}F(\gs^n),$$
where $\gs^n$ denotes the topological space given by the $n$-dimensional simplex.
In our case, $F(X)$ is the nerve of the category of principal $G$-bundles over $X$ and their isomorphisms.
Since any principal $G$-bundle over $\gs^n$ is trivial, we have $\def\C{{\rm C}}F(\gs^n)≃\B(\C(X,G))$, where $\C(X,G)$ denotes the group of continuous functions $X→G$.
Now
$$\hocolim_{n∈Δ^\op}F(\gs^n)≃\hocolim_{n∈Δ^\op}\B(\C(\gs^n,G)).$$
Since the functor $\B$ is homotopy cocontinuous, we have
$$\hocolim_{n∈Δ^\op}\B(\C(\gs^n,G))≃\B(\hocolim_{n∈Δ^\op}\C(\gs^n,G))≃\B(\Sing(G)),$$
as desired, proving that numerable principal $G$-bundles over an arbitrary topological space $X$ are classified by the simplicial set $\B(\Sing(G))$
(or the topological space $\B G$).
(This computation appears as Example 8.6 in the cited paper.)
It is easy to construct counterexamples already for the full subcategory $\{[0],[1]\}⊂Δ$. The category $\cal C$ can be constructed by applying the nerve functor to hom-objects of a category $D$ enriched in groupoids.
All groupoids have a finite set of objects and at most one morphism between any pair of objects.
The category $\def\id{{\rm id}}D$ has two objects $0$, $1$
and nonidentity morphisms $$s_0:0→1, \quad d_0,d_1:1→0, \quad s_0d_0=s_0d_1=\id_1:1→1, \quad d_0s_0≅d_1s_0≅\id_0:0→0.$$
The latter three 1-morphisms are isomorphic to each other.
However, $d_0$ and $d_1$ are not isomorphic to each other.
Now we have an obvious functor $N Δ_{≤1} → N D$ of quasicategories. Observe that the definition of the homotopy coherent nerve functor allows for an isomorphism $d_0s_0≅d_1s_0$.
On the other hand, a simplicial object $Δ→D$ does not allow for such an isomorphism.
Since any object weakly equivalent to the one constructed above must have $d_0$ and $d_1$ as the images of the corresponding face maps, we see that it is impossible to rectify the homotopy coherent simplicial object to a strict one.
Best Answer
The simplicial sets $h_2(N^\Delta(\mathcal{C}))$ and $N^D(H_2(\mathcal{C}))$ are isomorphic. To prove this, observe that the universal property of $h_2(N^\Delta(\mathcal{C}))$ applied to the image under $N^{\Delta}$ of the quotient simplicial functor $\mathcal{C} \to H_2(\mathcal{C})$ yields a map of simplicial sets $h_2(N^\Delta(\mathcal{C})) \to N^D(H_2(\mathcal{C}))$, which one can easily check is an isomorphism of simplicial sets (it suffices to check that it's a bijection on 0-, 1-, and 2-simplices).
The strict (2,1)-categories $|h_2(N^\Delta(\mathcal{C}))|^{\mathcal{D}}$ and $H_2(\mathcal{C})$ are biequivalent; in fact, there is a strict 2-functor $|h_2(N^\Delta(\mathcal{C}))|^{\mathcal{D}} \to H_2(\mathcal{C})$ which is a bijective-on-objects biequivalence. To prove this, note that the composite of the Duskin nerve functor (for strict (2,1)-categories) with its left adjoint sends a strict (2,1)-category $\mathcal{A}$ to its "normal pseudofunctor classifier" $Q\mathcal{A}$, which is a strict (2,1)-category with the universal property that strict 2-functors $Q\mathcal{A} \to \mathcal{B}$ are in natural bijection with normal pseudofunctors $\mathcal{A} \to \mathcal{B}$, for $\mathcal{B}$ a strict (2,1)-category. Moreover, by this universal property, there is a "counit" strict 2-functor $Q\mathcal{A} \to \mathcal{A}$ which one can show is bijective on objects and an equivalence on hom-categories, and hence a biequivalence.
The strict (2,1)-categories $|h_2(C)|^D$ and $H_2(R(|C|^C))$ are isomorphic (if you make a good choice of $R$, e.g. change-of-base along $Ex^\infty$). Indeed, the two functors $|h_2(-)|^D$ and $H_2(|-|^C)$ are naturally isomorphic, and the functor $H_2$ sends the "unit" map $\mathcal{E} \to R(\mathcal{E})$ to an isomorphism (for a good choice of $R$ as above).
I don't know any references for these answers, but these are all straightforward and standard arguments. If you would like me to elaborate on any of these points, I would be happy to.