> it would take nearly 1000 years to reach near zero

Depending on distribution, the rate might accelerate as it becomes difficult to find viable partners. You'd also experience civilisation collapse at a much earlier stage, as we've seen how COVID affected supply chains and created a new push for regional independence and de-globalisation.

I think war caused by conflict for resources is a much more probably cause of societal collapse, and indeed extinction.

Lower numbers of people in the world reduces the chances of that.

Birth rates are much easier to reverse than climate change - it's a really odd thing to focus on.

Area's of the world where birth rate is still high tend to be in poorly developed regions where human labour is required to survive.

However, in developed countries - the idea that we need the same or more people to maintain civilisation when the means of production is being increasingly automated doesn't make sense.

As an example ~100 years ago in Europe you were looking at nearly 50% of the population being involved in food production, whereas now it's about 5% - and that includes a 50% increase in the population to feed.

Note the EU is a net food exporter - even if it does import bananas.