The center of the enveloping algebra is polynomial on rank-many generators

By Harish-Chandra isomorphism for the center, the center $Z(U(\mathfrak{g}))$ is isomorphic to the Weyl-invariant polynomial algebra $S(\mathfrak{h}{)}^W$.

The Weyl group is a finite real reflection group on the rank-$r$ space ${\mathfrak{h}}^{\ast }$, so the Shephard-Todd-Chevalley theorem gives $S(\mathfrak{h}{)}^W\cong \mathbb{C}[f_1,\dots ,f_r]$ for homogeneous algebraically independent generators. Thus step 1.1 already proves the polynomial-algebra assertion.

To obtain the filtration degrees directly, use the PBW symmetrization map $\mathrm{sym}(x_1\cdots x_m):=\frac{1}{m!}{\sum }_{\sigma \in S_m}{x}_{\sigma (1)}\cdots x_{\sigma (m)}\mathrm{.}$ The PBW basis in PBW gives an ordered monomial basis for the enveloping algebra makes this a filtration-preserving vector-space isomorphism whose associated graded map is the identity. The adjoint action is a derivation on both algebras, so $\mathrm{sym}$ is $\mathfrak{g}$-equivariant. It therefore restricts to a filtered vector-space isomorphism $S(\mathfrak{g}{)}^{\mathfrak{g}}\stackrel{\sim }{\to }U(\mathfrak{g}{)}^{\mathfrak{g}}=Z(U(\mathfrak{g})),$ and gives $\mathrm{gr}Z(U(\mathfrak{g}))=S(\mathfrak{g}{)}^{\mathfrak{g}}$. Chevalley restriction for symmetric invariants now lifts the homogeneous $f_i$ from step 2.1 to homogeneous algebraically independent $p_i\in S(\mathfrak{g}{)}^{\mathfrak{g}}$ of the same degrees.

Put $z_i:=\mathrm{sym}(p_i)$. Then $z_i$ is central, has PBW filtration degree $\mathrm{deg}{p}_i=\mathrm{deg}{f}_i$, and has leading symbol $p_i$. Since the $p_i$ generate $\mathrm{gr}Z(U(\mathfrak{g}))$, subtracting a polynomial in the $z_i$ with the same leading symbol lowers the degree of any central element; induction on PBW degree shows that the $z_i$ generate the center. A polynomial relation among them would give, in its highest filtered degree, a relation among the algebraically independent $p_i$, so no such relation exists. Hence the $z_i$ are algebraically independent generators with the asserted definite PBW filtration degrees.