To me the question itself (and the answers) are out of focus, starting with the claim  that the ring of Weyl group invariants is somehow central. Chevalley's 1955 argument does show that this ring is a polynomial ring, generalizing the classical ring generated by elementary symmetric polynomials (equal in number to the rank of the derived group).    But the extra argunents by Harish-Chandra to realize the center of the universal enveloping algebra of the Lie algebra involve a more subtke $\rho$-shift.   (There are other arguments needed in prime characteristic.)

It's helpful here to avoid the magic word *reductive*, since the main issue is the *semisimple* case.   This group always has a fibite center, trivial if the group is of adjoint type.     In general a reductive group is the almost-direct product of a central torus and a connected semisimple group.