I think the important point was not, "If this happens, it could destroy the world," so much as "This may already have happened. Look how easily it almost happened just now. And we'd never know until it was far, far too late to even save ourselves." The harping on what might happen isn't scary in itself. (Also, I don't recall them going on about it overmuch, but I was a little slower to catch on to what was actually going on, as hard SF isn't really my genre.)
It's the old nervousness about nuclear annihilation with an interesting wrinkle to it.