I think the EFF is dead wrong here, but I'm open to be convinced otherwise. Aside from the trappings of fame itself, is there any precedent where a public official has been restricted from certain rights that they previously enjoyed in their private lives?

Maybe once you become an elected official, you can no longer chase people off your front lawn. They have a right to speak so maybe that's more important than your private property rights. Maybe they can demand you charge their iphones while they're protesting on your street corner? The government is paying for their salary anyway, shouldn't a protested politician be required to support the protestors in the name of free speech? How far does this go?