No it does not, not for countries like the US that are primarily composed of immigrants. I think we often forget that a lot of the white people here are immigrants, too, usually only a couple generations removed.
Presumably either the countries being emigrating from must be at replacement rate, or themselves declining? Somebody has to be producing the next generation, somewhere..
But the point is that the illness of developed nations is having the symptoms treated by infusions of immigrants from countries that don't _yet_ have the same illness. If it's a matter of "as countries develop they have less children", you'd better hope other places of the world stay poor and uneducated.
As rights for women progress, they choose not to have children. Which makes a lot of sense - having children is risky. The reason women in underdeveloped countries choose to have them is because they have less rights and less autonomy, as well as a worse economy.
If you're poor and require a husband to function, then having children is a huge boon. Not that you have much of a choice - your husband usually chooses. He bears little to no risk to having children, so of course he says yes.
What we cannot do in the West is backtrack on education and equality. It would probably work, yes, but we would literally develop backwards. Our economy relies on women working, our economy relies on education, etc.
So, instead, we should make having children a sweeter deal, rather than pseudo-forcing women to have children. We should offer generous and safe maternity leave, we should offer free childcare, and we should implement universal healthcare to eliminate risks associated with infant care, pregnancy, and postpartum care. We could even make higher-education free, to eliminate risks associated with saving for college.
Basically, women aren't having children because it's incredibly risky for little to no gain. By having children, you are ensuring greater economic hardship, health challenges, and less autonomy. It's a Catch 22 - if you choose to stay home, you have sabotaged your career. If you choose to work, you must pay exorbitant childcare fees.
We should not be surprised that women are opting out.
We live in such unprecedented times I'm again unsure why you would choose population growth rate as a factor to single out as particularly interesting and not, say, wealth inequality, climate stability, relative access to basic resources goods and services, perception of the competence of the ruling class, etc etc
The total number of people living in the world does not matter to local areas that see themselves age rapidly and hollow out as young people leave and they become unable to support the generous welfare we give to the old.
In other words: it's quite famous for how absurdly enormous swings in birth rate can be. It's famous for how critical it is for a species to have a stable birth rate.
I don't know where you got that idea. Some species critically depend on wildly unstable birth rates (grasshoppers and cicadas, but probably also deer and many other prey populations).
Stable populations are completely irrelevant at the microscopic levels; InBev would fold within a week if yeast populations were stable.
And both grasshoppers and cicadas are famous for suddenly disappearing across enormous areas. Which is a situation only very rarely referred to as "stable".
I know there's a joke in here about this being literally in the bible, with God using such an insect birth rate swing as a punishment for an entire state. That's how "stable" it is.