Let $D_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of parenthesized braids. An object of the groupoid $\PaB(n)$ is a complete parenthesization of a permutation of the symbols $\{1,...,n\}$, e.g. $$ ((24)(13)5) \in \mathrm{ob} \,\PaB(5),$$ and morphisms are $n$-strand braids, such that the start- and endpoints of each strand are labeled by the same element of $\{1,...,n\}$. Composition in this operad is given by "cabling". The connection to the little disk operad is that one can quite easily write down a morphism of operads $$ \PaB \to \Pi_1(D_2),$$ where $\Pi_1(-)$ denotes the fundamental groupoid, such that $\PaB(n) \to \Pi_1(D_2)(n)$ is an equivalence of categories for all $n$.
If we denote by $N$ the composition of the nerve functor with the geometric realization, then $N\PaB$ is a topological operad.
Fact: $N\PaB$ is equivalent to $D_2$, i.e. there is a zig-zag of operad morphisms between $N\PaB$ and $D_2$, each of which is arity-wise a weak equivalence.
In the literature this fact is in several places proven by appealing to Fiedorowicz's recognition principle, see e.g. Dmitry Tamarkin, Formality of chain operad of little discs, Section 2. I don't understand why and I guess I am missing something: it seems to me that there is a much more direct argument. Indeed the map $\PaB \to \Pi_1(D_2)$ gives a homotopy equivalence $$N\PaB \to N\Pi_1(D_2),$$ moreover, the unit of the adjunction between $\Pi_1$ and $N$ gives a morphism $$D_2 \to N\Pi_1(D_2)$$ which is also an equivalence since $X \to N\Pi_1(X)$ is a homotopy equivalence for any $K(\pi,1)$-space, such as $D_2(n)$. What am I missing?
