Reality check. We are in ZERO danger of extinction. Our population could decline by 95%, and there would still be more humans alive than when Julius Caesar walked the Earth. Extinction would take millennia. But even then, there are built in safe guards that make declining below a certain population level impossible.
For a start, consider natural selection. societies are not homogeneous. Some groups reproduce at above replacement, some below. In the US for example, if we just assume birth rates remain the same for all groups indefinitely, never returning to an average replacement rate? In that unlikely scenario, extinction still will not happen. The Amish and the Mormons shall inherit the Earth.
And this can happen for cultural practices, not just religious groups. Maybe multi-generational households will become more common through natural selection. Maybe it will become the norm for people to have children right after high school, while still living with their parents. You only go to college and start your career after having some kids. Each generation pays to raise their own grandchildren, rather than their own children.
And even without cultural selection, eventually technological decline will prevent extinction. What technology your society can wield is a function of its population. As your population shrinks, your society's technological complexity decreases. Eventually you can no longer manufacture mechanized farming equipment, so more people have to work the land again. And on traditional family farms, children are an economic resource that pay for themselves. So the birth rate recovers. And of course, reliable birth control is the product of industrial production. Eventually your population shrinks enough that you are no longer capable of running a birth control factory.
I don't want to see the country become entirely Amish or to see us collapse back to 18th century levels of technology. But these are just the worst case scenarios. The absolute worse case scenario is that population declines by 90-95% and we regress technologically to some 18th century equivalent. Not a future I want to see, but it would hardly be the end of the world. Our present industrial civilization would be remembered as an unsustainable historical aberration, a temporary population boom brought on by the era of cheap hydrocarbons.