With all due respect, "what's not being said" -- silence, etc. is part of "dialogue."
With all due respect, it's not. *Shakes fist at maestro*
Dialogue, by etymology, is the spoken part of conversation. "Not talking" is still communicating, but not through dialogue.
At least that's my definition.
Indeed.
Perhaps I've been hanging around novelists for too long.
Perhaps. I think, though, that discussions such as this get mired down in
personal "definitions" of words, and artistic
opinions of concepts, rather than adhering to the actual meanings of these concepts. It's very artistic and imaginative and warm and fuzzy to consider silence as "dialogue" and to embrace and romanticize that idea as some truth, and may even make the author of that book seem somehow insightful (though misinformed), but it doesn't actually fit the defintion of what dialogue actually is. (Dialogue from
dialogos where dia means "through or across" and
logos means "speech, word, reason." Also akin to Greek
legein "to gather, say." Also
dialegein "to gather and speak").
Silence is part of a conversation. It is part of the communication of ideas. However, it is the non-spoken, non-dialogue, part of the overall conversation. This is especially true in terms of screenwriting, where dialogue (usually) strictly means the spoken words of the characters.
And, I think you've hit on another great point. Dialogue, by itself, doesn't necessarily make a great conversation between characters. It is the interplay of the dialogue (words), facial expressions, overt actions, silence, etc.
(If you want to be technical, dialogue itself is an action. But, that would be too broadly inclusive a definition for this debate.)

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Back to the original question, I still believe that good structure can save bad dialogue, but good dialogue cannot save bad structure.
Dialogue comes from the characters. It is their reaction to the situations the writer puts them in. The situations themselves, however, are guided by structure.
Structure comes first. Then dialogue. That's the natural progression of a well-crafted story. That's why, to me, structure is more important (in your hypothetical scenario), though only slightly.
IMO, Screenplays evolve from a top-down process, IMO. Correcting things from the higher orders has an affect on the lower ones (trickle down). That's why, IMO, concentrating solely on dialogue to strengthen a story is the wrong approach, because that is a bottom-up process (you can't trickle up).
Adjusting structure will invariably have an effect on dialogue (as it will influence the situations and motivations of characters), but adjusting dialogue will not necessarily have any effect on structure.