If there a water pipe electrode from a common water system then the GEC may help with an open neutral. If there is just a ground rod or two then it probably won't do much to help the open neutral problem because it's relying on the dirt as a return path for the neutral current.
If it was a water pipe electrode that may be. A ground rod wouldn't make a bit of difference, the contact resistance with the dirt is too high. There is a reason that GEC to a ground rod never has to be bigger than #6.I've had open neutrals and bad neutral (not totally failed), and this would give significant current and voltage onto the GEC at the first panel that bonding is made. It would seem this gave some measure of safety that otherwise wouldn't be there absent a GEC connection.
That would be the lack of an equipment ground.I've also had where the GEC was not made and the skin of the mobile home became energized with no path back so no breakers tripped but had 120V on the metal skin.