> 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.