Why not? I don't think it will happen, but at the same time, I don't want to say anything about human behaviour without stating assumptions. If you think (as do I) that there is always some point above zero population at which birth rates do stabilise, then it's valuable to examine what exactly that would entail.
It sounds like we agree the point exists somewhere - we could extend your example to the whole Earth's population becoming 2 and the population becoming (8 bn / 2)% more productive. Why can't we extend it to the population becoming 1?
It's a completely different situation, and none of the shit we are arguing about applies to it. There's a fundamental difference between running and meeting the long-term needs of a human society of 10 billion people and a human society of 10.
The mathematics you're using is the same reduction to absurdity that leads a manager to believe that nine women can deliver one baby in one month.
And worrying about 'what will happen if human population actually drops to zero' is like worrying about commuter traffic congestion on Mars.