Little girls raised by mothers who only have 1 child don't grow up thinking that they want five children. Little girls who grow up being taught by childless teachers, excited and inspired by childless role models in the entertainment industry don't think they want four children. When they're bombarded with propaganda about how "adoption is just as good", they tend to believe that. And so on.
These things don't change just because population is shrinking. I'm not a conspiracy theorist, I don't believe these things to be deliberate. But I do recognize them for what they are.
> There is no reason to expect that the population will die out in a few century, only that we are in a currently bad situation.
There actually is. People like yourself tend to look at the "population". "Gee, there's 7.2 billion sixty year olds! We'll never go extinct." But it's really the one small demographic slice that is shrinking all the time, even while population grows... the demographic still young enough to make more people.
Your claim is that this trend doesn't just stay the same, that it might reverse. But it's so much worse than it merely not reversing... the trend becomes more extreme. As fertility falls, there are far fewer examples of high fertility for children to grow up around and be normalized to. They lower their own fertility in adulthood in response. Fertility decline is set to accelerate, not slow.