The idea that Kennedy could have yanked Stevenson out of the session as though he were a pitcher who's lost his stuff is silly at best. The result is a pretty shameful example of audience manipulation. As is the scene where a U-2 pilot attempts some frantic maneuvers to avoid a missile. That's the problem with trying to make a movie out of a historical event. History does things its own way, without the structure we demand from our entertainment.
History has confusing subplots, poor dialogue, and a lack of focus.
Sometimes the heroes are hard to distinguish from the villains. Events usually don't follow the dramatic arc that we want from movies, and the ending isn't always a crowd pleaser. Filmmakers who take on historical subjects realize this and try to tailor the events so they better fit into the movie mold. Though in fact it's a distortion of a true story. The obvious answer is that they know good stories when they see them.