Once again. Slowly.
As people gain access to medicine and education, and as quality of life increased, traditions and religion fly out of the window. Even deeply religious countries like the Gulf states go from 7-8 kids per woman to roughly 2 kids per woman.
These factors trump any other factor in determining how many kids a woman will have.
Once the countries get on more or less the same level in terms of these factors, only then will you see variations due to other things like culture, religion, what not.
Israel is 1.5 times more than other developed countries (and the difference is slowly but surely dropping)? So? Niger is almost 4 times more than other countries.
Medicine, education, quality of life. These three universally affect rates regardless of tradition, religion or what not.
> On average most woman want two children. But when a lot of them die they will get more (average of 6 around 1800).
and that doesn't seem to be the case.
And you said that culture had nothing to do with it, but clearly the variation in the birth rates (50% to 150% of the ~2) in countries with roughly the same standard of living is due to something.