When you say you wait for everybody to stop arguing before pulling the thread that unravels the sweater, that's not Ni. It's how an introvert uses Te. (Edit - more accurately, it is how INTJ uses Te.)
As for "but having heard all of the arguments could create a model in his head of what was going on, the model being a synthesis of the thoughts of the people arguing," it could be Ni if the model was an idea or concept created subconsciously, as with my Poincaré example. You are trying to explain something mysterious using psychological terms to make plain what is not plain.
And that's why I favor using examples. Because the examples themselves are plain. I intuitively (viz, from experience) understand Jung dreaming about something represented metaphorically that came true. I don't however understand it. Calling it a synthesis doesn't explain it.
Ni is mysterious, thus this thread.