I don't think I ever heard that a deal had been reached with the owners of the original theatre, to allow Disney to use the theatre's likeness as a weenie.
My memory isn't always the greatest nor is my knowledge of all things theme park encyclopedic ... it could be that either the story about the weenie-licensing conflict has been exposed as a hoax, or perhaps a deal actually has been reached. But if the story still stands and a deal hasn't been reached, then the stage will pretty much HAVE TO hide the theatre.
I do know that in the field of photography, there is a growing legal chill against ever using ANYBODY'S identifiable building, structure, or any of their property as the main subject of any commercial photo without the written permission of the building/structure/property owner. Photo agencies therefore state that they will reject artists' submissions of commercial stock photos if there appears to be anyone's identifiable property in the photo.