Per the OP -How is the voltage stabilized? Long answer short via the GEC. For the whole picture as to how that happens, reading the whole post-
Is it not a given that under normal conditions the grounding conductor would be the same potential as the earth?
I agree with Bob for the most part. However, we have funny things happen in the world. Where there is opportunity for charges to build on oposite sides of the "plates", grounding insures that there is "zero potential" between the plates, based on the principles laid out in 250.4(A)(1) It's about
incidental contact of high voltages. I consider grounding, and bonding two seperate, but related, issues.
In 250.24(A)(2) and In 250.24(A)(3) the BIG difference is that in (3)
Shall be connected to earth so as to limit the voltage to ground on these materials VS
shall be connected together and to the electrical supply source in a manner that establishes an effective ground fault current path.
In 250.24(A)(2) is to drain (limit) into the earth (via the GEC per 250.24(D)), to balance (create zero potential) any voltage imposed per (A)(1).
In 250.24(A)(3) Is normally to interrupt the OCPD to stop current flow to protect life or property.