I'm an example: I wanted a large amount of children, and taught my children the joys of a larger family. So far, they think along the same lines.

I don't think one can determine if the trait is or isn't cultural by guessing. How could you possibly know the origin of your own mental preferences? Even political affiliation, which sure looks cultural at first glance, has been shown to be under genetic influence.

There is evidence that there are certain genes which promote religiosity. They don't guarantee it – somewhere out there, there is probably an atheist with lots of religiosity promoting genes, and a devout believer with none of them – but they do make religious belief more likely.

People who have large families for religious reasons will pass on both their religious beliefs to their children, and also in many cases their genetic predispositions to greater religiosity. Of course, there is no guarantee that the child will stay in the parent's religion, but so long as the retention rate is decent (say >= 70%) the religious belief (and the genetic predisposition to adopt it) will be spread widely through the population.

Even though organised religion isn't looking very healthy at the moment (at least in developed countries), this is reason to believe that its long-term prospects are actually quite bright.

It's well accepted that many childless women in their 30s will start to crave having children. Someone with a mutation that causes this craving to happen in their 20s as well will have more kids on average.