One of the strange things is that the Soviets had eventually developed something similar as described in the movie. It was as automated as in the movie but the general idea is the same

https://en.wikipedia.org/wiki/Dead_Hand also called Perimeter (Периметр).

There is an interesting point in the movie that the idea was to always inform the opponent about it, so they don't even dare strike first. But the system was kept secret, at least officially, for many years. One of the designers explained that the other purpose of the system was to protect against hot-headed generals who, on short notice or from a faulty warning signal, may panic and launch an attack. In that case it wasn't necessary to advertise the system.

My idea is that the opponent (US) also knew about it from intelligence channels, and the Soviet knew that US knew, and that was enough. There was no point in officially making it public.