Why is acidic cooking problematic for cast iron pans but not BBQ plates?
Iron is a very reactive metal, responding to many different influences. Heat, acidity, moisture, even the oxygen content of its environs cause reactions.  Barbecue plates are manufactured with enough other metals (zinc, aluminium, etc.) and via a tempering process that keeps the metal chemically stable in reactive environments. Cast iron is usually pure iron, and it just follows its nature when provided with a reactive environment.