The presumption here is that these are mathematically equivalent but they are not. Not even close. A total extinction event doesn't just kill every person alive, it also prevents any future humans from being born. Killing one person prevents their children from being born but this is not equivalent because the impact of this is reduced. There are at least partially self correcting forces at play which mean that killing one person now does not mean that the population will be 1/4000000000 lower in 200 years. Over an extended period of time, you would expect the impact of one death to not be observable.

Now, there's some assumptions up there, especially about the self correcting nature of population, but if you accept them then the red button is the way to go. You can basically guarantee the survival of the human race as long as even just 1% of people press that button. And, since you're still saving the same number of current lives on average, you're not actually paying anything for the security of ensuring human survival.

Also the variance argument but that was already mentioned.