Ultimately, this talk of humans dying out (and the supposed merits of that) is childish.
(Well, childish is perhaps a little generous).
It's not realistically going to happen, not from climate change and not from the fertility crisis.
At warming of 3°C or 4°C, before 2100, there would be a risk of widespead societal collapse.
At 5-6°C, large areas of the planet would become all-but-uninhabitable, but extinction would remain unlikely.
Above 6°C, the theoretical risk of extinction becomes a real one.
Even if we simply continue on with current policies, to the extent they have already been implemented, the IPCC projects warming of 2.5-2.9°C.
We should absolutely be striving to do far better, but no serious person thinks extinction is even close to being on tbe agenda from climate change.
While I'm sure everybody recognizes that a drop in the global population is desirable, the demographic challenges it brings will be extremely difficult to navigate, and is something that should be confronted at the earliest opportunity to offset the worst of it. That could be by trying to manage the pace of the decline and/or through relocation of resources (including new technologies).
I suppose, though, that making ill-informed, unscientific declarations of how we're all doomed is a nice and convenient way to absolve yourself from feeling any responsibility towards any generation that is younger than you - particularly if it involves even a modicum of personal sacrifice.
Its the same brand of selfish ignorance, greed and science denialism that will almost certainly see humanity miss the best-case scenario of warming of around 1°C, but with a cherry of hypocrisy atop it.