I'm not convinced we can. For starters, the nuclear weapon inventory is a lot smaller than it was back in the 80s when those sorts of statements were popular. And secondly, as I understand it, it was always a bit of a lie, an estimate based on some silly assumptions (e.g. that everybody in the world lives in cities), which in repetition eventually lost those assumptions.
And yes, I agree with the other poster that it's pretty much impossible to kill everyone on earth with climate change.
Species-ending diseases I do worry about, but maybe that just means I don't know as much about immunology as I do about nuclear weapons.