> It’s true that birth rates must eventually flatten out and become sigmoid

All positive growth eventually flattens out and becomes sigmoid, but a lot of phenomena experience negative growth and nose dive. No gentle curve, but a hard kink and perfect flat line at zero. Forever. I think it would be a stretch to categorize that pattern as sigmoid. Predicting a sigmoid pattern for negative growth implies some sort of a soft landing (depending on your definition of soft).

We can think of many populations that are no longer with us. So just a caution about over applying this reasoning in the negative case.