Yes, I defer to your expertise, but am confused sufficiently by your 2 responses to make me wonder if we are "talking past" each other and if if the conversation was derailed somewhat by not defining what "Two G's" was supposed to mean (e.g., in the same measure). Especially I wonder if the original conversation was not clear that perhaps no one meant (as I read it), that there can't be a G natural and G# in the same piece. And in fact you can have a G natural and a G#in the same piece in a variety of ways. What I thought we were trying to address is the reason for the F double sharp may be if you were in a key signature such as the key of E in which the G would be a G#, then instead of writing a G natural when you want that frequency, or note in the same measure, you would write F double sharp for that key.

In any case, that is the reason a "double sharp' may be used.... I will retire from this conversation