Excellent essay. Though I must admit I've always found Dr Strangelove a bit too broad to be effective. Why, for example, did Kubrick insist on Scott's Buck Turgidson be played in such a burlesque manner? The story goes that Scott strongly disagreed with that portrayal, that Kubrick finally persuaded him to play it that way just once, for reheasal purposes, secretly filming & putting it into the final cut. For which Scott never forgave him.
I don't know if that's true, or even plausible. But I think Scott was right. Why would we be frightened at such a cartoon character? It draws the viewer out of the story, reduces verisimilitude.
I think the same is true of many of the other characters.
That said, Hayden's speech, that starts by quoting Clemenseau, so rational descending into utter madness, is one of the great moments of cinema. And, as you point out, the set design is nothing short of brilliant.
Then there was, I guess, the happy accident of Sellers not being able to portray Major Kong & being replaced by Slim Pickens ... he was just brilliant. That scene with him riding the H bomb down, another great moment of cinema.