I really hope that line would have been "the weapons are good enough", but considering the US stockpile of nuclear weapons at the end of the Cold War was enough to cause the extinction of humanity a few times over, I guess not...
The claim that we have (or had) enough nuclear weapons to cause the actual extinction of humanity, much less cause it several times over, is dubious. If you google it there are numerous articles debating it. At the very least there isn’t scientific consensus.