We were still watching that video in elementary school in the '70s. By that time, I think we all understood that if a nuclear bomb landed directly on us, we were done. As I recall, "Duck and Cover," was meant to minimize the impact for people who were lucky enough to be far enough away from the impact to survive the immediate blast. We even had lessons that went along with it where the teacher drew circles and described the expected effects at various distances from the blast, and she told us the distances where they thought you'd be able to survive by shielding yourself. I think the bit about the car crash was just to put it in terms that a 2nd or 3rd grader could relate to.
Then in the eighties, we talked about, The Day After in school for what seems like a week or more.
I vividly remember watching The Day After with my family. I think just about everyone I know who was born in the 1970s watched that movie. It was quite different when there were only 4 networks (possibly only 3 at that point). There were a lot more universally shared entertainment experiences.