If that's the case, without a radical cultural change, humans are basically going extinct.
This is definitely not true. Look at the fertility rates of any of the aforementioned cultures. Humans will never go extinct because they're making the conscious choice to do so...
This clearly isn't true because the Amish and Ultra-orthodox Jews still want/have 7-8 kids, and they live in modern society. I'm not sure about the Ultra-orthodox, but the Amish are certainly capable of sustaining their standards of living even if the rest of America dies out, so humanity wouldn't go extinct, it'd just become 100% Amish (and whatever other cultures are able to maintain persistently high fertility rates).
Give Amish and Ultra Orthodox Jewish women financial independence and see how many kids they have.
If you include the option of extinction, sure.