My question is: Does the vanishing of the Poisson bracket plays an important role in finding and proving Duflo's isomorphism theorem? Or it is just an literally first step?
Let $A_0$ be a Poisson algebra and $A$ a deformation quantization of $A_0$ (assume we are in a context when it exists).
Assume you have a quantization map $Q:A_0\to A$, by which I mean a section of the classical limit map $A\to A/(\hbar)=A_0$.
Then for any two elements $a,b\in A_0$, $[Q(a),Q(b)]=\hbar\{a,b\}+O(\hbar^2)$.
Hence if you want to have $Q(ab)=Q(a)Q(b)$ you must at least assume that $\{a,b\}=0$.
My (non-)answer to your question is then:
the vanishing of the Poisson bracket is a necessary requirement if you want a statement of Duflo-type. It is just a first step.
The actual history comes from the Harish-Chandra isomomorphism. Duflo noticed that the original formula could be written for any Lie algebra, without any use of roots and similar stuff specific to the semi-simple case.