Would it not be really dysfunctional for cultures to compete by producing more babies than their opponents?
I am not so sure.
For one thing it depends on whether the culture is living in a closed system, independently of others, or is a minority in a larger culture. There are regions in the world where some minority groups are intentionally breeding at a faster rate to achieve greater power vis-a-vis the majority. They have already succeeded in this in many contexts.
Population increase is for some groups (religious, racial, or linguistic) what the bottom-line is for capitalist investors. They give a damn for what happens to the world at large, or for the majority of the nation of which they are a part. Their goal is to fatten their purse because this gives them more power with respect to others.