The center of the universal enveloping algebra of a semisimple Lie algebra (say over $\mathbb{C}$) $\mathfrak{g}$ is known to be a finite rank polynomial algebra. The justification I know involves the Harish-Chandra isomorphism $S(\mathfrak{g})^\mathfrak{g}\simeq S(\mathfrak{h})^W$ where $\mathfrak h$ is a Cartan subalgebra and $W$ the Weyl group. The Chevalley-Shephard-Todd theorem states exactly than $S(\mathfrak{h})^W$ is a polynomial algebra. We also have the degree of the homogeneous generators of the invariant algebra [1].
A set of free generators of $Z(\mathcal{U}(\mathfrak{g}))\simeq S(\mathfrak{g})^\mathfrak{g}$ is of course non-unique, but the two usual presentations I see are given by traces of powers or by elementary symmetric polynomials, both in the adjoint representation (see for example here). These being polynomials on $\mathfrak{g}$, we obtain elements of $S(\mathfrak{g})$ via the ($ad$-equivariant) Killing form.
They both generate the same algebra, that is the polynomials on $\mathfrak{g}$ that are invariant under any linear automorphisms (or infinitesimal automorphisms), this is for example proved here. The corresponding polynomials are then necessarily invariant under the adjoint action of $\mathfrak{g}$.
My question: Can we directly justify that they generate the whole $S(\mathfrak{g}^*)^{\mathfrak g}$ algebra? I have seen in several places (like this article or Wikipedia) that it is a consequence of the Poincaré-Birkhoff-Witt theorem but I don't see how.
[1] : Chevalley, Invariants of Finite Groups Generated by Reflections
