My favourite book of the past few years, Eric Kaufmann's "Shall the religious inherit the earth", points out that some religious minorities, often quite extreme ones, are much better at having children, and are likely to spread at the expense of the mainstream. This cultural difference in family stability reminds me of that.
I would like to believe that the US and Europe can pick up whatever enables poor immigrant families to maintain stable families and viable reproductive structures – without buying into the more extreme aspects of some pro-natalist religious cultures.
It depends how cultish and restrictive the parents are. I know of some cults that even handicap their kids' English skills and keep them mingling amongst their own so that they don't really have any outside resources. Once they're economically dependent on their tribe, it's much harder for them to break out.
Yes, but those children that move away also move towards Western norms of sub-replacement fertility; while the children that don't, don't.
I wouldn't say the best way to think about this is "these crazies are baby-making machines!" (Strawmanning your argument a little there.) There is some craziness in e.g. Quiverfull protestantism or Ultra-Orthodox Judaism, sure. Nevertheless, if culture X is set to survive into the future on current trends, and culture Y is not, then maybe culture X isn't completely stupid.
I cannot find the citation anymore, but I remember reading about an interesting phenomenon related to this. Since I cannot find the source now, please call me out if it is actually wrong, but it seems to make sense and rings true to my personal anecdotal experiences.
Basically, the hypothesis is that immigrant families themselves indeed move away from extremism. But their children born in the new country tend to overcorrect and try to "discover the roots" by swinging harder in the opposite (extremism) direction. One generation down from the one that became less extremist, basically.
I might be confusing some details given how long ago I had read it (it might be not the children of the immigrant family, but children of their children, for example), but that's the gist, and I saw it happening myself more than once.
I would like to believe that the US and Europe can pick up whatever enables poor immigrant families to maintain stable families and viable reproductive structures – without buying into the more extreme aspects of some pro-natalist religious cultures.