This is incorrect. Winestock:
> Bottom line: Lack of religion is either non-adaptive from a fitness perspective, or it is strongly tied to traits which are themselves non-adaptive.
A third possibility, one of many, is "Lack of religion and a fertility rate less than 2.0 occur co-incidentally in seven countries". Given that we have only examined nine countries of a possible 195 (or so), I think that this third option is at least possible.
> Lack of religion is either non-adaptive from a fitness perspective, or it is strongly tied to traits which are themselves non-adaptive.
There is at least one other possibility, which is that there is no link between religion and fertility, even transitively.
If we find a correlation, then that's a sign that we should look deeper. So far, the evidence says that lack of religion is tightly connected with a drop in fertility. If something other than lack of religion is causing the drop then the best place to look for it is by checking those factors that are most common to the irreligious and least common among the religious.
If someone has done this research then I am unaware of it.
I might add that for social causation issues like this, longitudinal data (studies of changes over time) are often much more informative than cross-sectional data (comparisons of different subgroups at the same time). You and the original submitter posted longitudinal data, studied by researchers who have a data model that they are further investigating. That's at least worth talking about here.
After edit: as several replies have helpfully noted, one way to test the proposed model is to look at data from other countries, and another way to test it would be to look at data from different (earlier) historical periods in the same or different countries. Much work needs to be done to accept the causal mechanism proposed in the submitted link, but meanwhile we can check the narrower issue of whether participation in organized religion has the same time trend (reduction over time) as fertility rates in various countries of interest.
The same model would probably fit quite well to countries where there's a marked trend increase in a specific religious affiliation, like Indonesia where dominant and growing Islam may hypothetically be decades away from making other affiliations extinct. Does Indonesia have a correspondingly much higher fertility rate? Actually no (~2.28), especially not if you control for other variables like poverty and infant mortality that are known to be correlated with high fertility rates. And it's dropping.
Correlation != causation is laziness at it finest.