Nukes just... work. You twist a couple of keys, flip a switch and 45 minutes later human civilization ends.
A weaponised virus doesn't have these constraints, and could destroy humanity, but at least it takes some time, and can probably be somewhat mitigated.
In nuclear war, a split second decision could annihilate every living human, erase all evidence of our existence, and maybe cause a big enough climate shift to permanently end life on this planet.
Not to take too much from the argument, but that line is way overblown. I think it's hard to believe a nuclear war would kill all humans (it would certainly destroy civilization, but killing everybody is way harder) but nobody can be sure of that. Anyway, the Earth has survived meteor strikes larger than humanity's entire arsenal and life is still here.
Biological weapons can probably do this 10x faster and via weaponized viruses meaning no ICBM's needed, just a good enough virus to be carried by human and animal hosts that infect the world. Or infect common crops and watch the world burn as everyone starves.
A total war between the US/NATO and USSR sans nuclear weapons would also probably have ended western civilization and probably drawn in everyone else too. World-killer? Maybe, maybe not, but the world left over would have been one no one wold want to live in. The same way a post-nuclear world would be.
If anything nukes kept US/NATO and the USSR from going hot. That conflict could very much been a world-ender and was itching to happen for decades.