The stabilization of voltage with respect to earth is a real feature of ground rods, but it is limited by the ground rod resistance. To the extent that the perturbing current source is very high impedance, the ground rod will keep the electrical system voltages referenced to earth, but drop a high current source on to the electrical system, and the ground rod will act in exactly the fashion that you describe; the electrical system voltage will go someplace strange, and the ground rod will take a small patch of earth with it.
Static electricity (which you agree ground rods 'bleed off') is such a high impedance source. The capacitive coupling that causes 'phantom voltage' is also a high impedance source. Minor insulation leakage current is a high impedance source. All of these things could charge bonded metal relative to ground, but are of such high impedance (low currents) that a ground rod could actually maintain a good earth reference.
-Jon