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 $[n]↦(\Sing(G))^n$ because $\N$ is induced by the homotopy cocontinuous functor from simplicial spaces to spaces that sends representable simplices to contractible spaces. 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.)

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.)