Today those countries have some of the lowest birth rates in Europe, e.g. Spain and Italy[1] pretty much trail the pack. The data suggests that your hypothesis is false.
1. https://en.wikipedia.org/wiki/Demographics_of_the_European_U...
High birth rates are an important aspect of Orthodox demographics: They are the most reproductive of all Jews, and ultra-Orthodox communities have some of the highest rates in the world, with 6 children per an average household. Non-existent levels of intermarriage (unlike some liberal Jewish denominations, Orthodoxy vehemently opposes the phenomenon) also contribute to their growing share in the world's Jewish population. While American Orthodox are but 10% of all Jews, among children, their share rises immensely: An estimated 61% of Jewish children in New York belong to Orthodox households, 49% to ultra-Orthodox. Similar patterns are observed in Britain and other countries. With present trends sustained, Orthodox Jews are projected to numerically dominate British Jewry by 2031, and American Jewry by 2058.
However, their growth is balanced by large numbers of members leaving their communities and observant lifestyle. Among the 2013 PEW respondents, 17% of those under 30 who were raised Orthodox disaffiliated (in earlier generations, this trend was far more prevalent, and 77% of those over 65 left). It is estimated that over 20% of those raised religious Zionist in Israel disaffiliate, and greater numbers adopt a secularized lifestyle and define themselves as "living on the spectrum (of religion)". Loose observance among young adults is common even when they retain formal affiliation.
And should big families in Catholic-majority countries 100-200 years ago be attributed to Catholicism? Well, other countries, with completely different religious beliefs (e.g. Muslim-majority countries, India, China) also had big families. Most people back then, whatever their religion was, had big families due to primitive contraceptive technology, high infant mortality, economic value of child labour, less years spent in education, lack of educational and career opportunities for women, etc. Nowadays, there is a correlation between religiosity and family size, but that link is historically novel; for most of human history, the devout and the non-devout had big families alike.
It's interesting that religion has remained relevant for many whose lives today match what you describe. This isn't to blame religion for those problems, but it doubtless plays a role as both cause and effect, especially when the state is unable or unwilling to provide an adequate level of life security for the populace.
With the growing educational opportunity of catholic women, many decided to not have large families. Many people believe that this trend will continue. In my experience however it will not. The traditionalist Catholics tend to be more highly educated than their laxer counterparts. The women are educated but have rejected their mothers desire to put work ahead of family.
I think the sudden decline in catholic family size is mainly due to the suddenness of the changes to educational possibility for men and women and the rapid industrialization the world experienced in the past century. However we're at the tail end of that and educated catholicd are no longer making the same choices as their parents. I think this demographic effect will be a short blip in the history of the church.
Speaking from my own experience as a rather traditional catholic (we dont use contraception and dont plan on it which puts us firmly in the crazy category according to most), all our friends and their wives who are also forgoing contraception and are planning for large families are highly educated. We are graduates from top schools. Mothers often have advanced degrees in stem subjects. Others are actually working towards their doctorates. Still want kids and as many as God gives.
This is my experience in the san Francisco bay area and the fact that there are so many traditional catholics here in the knowledge sector despite the bay areas reputation of ireligiosity is telling. Catholics are quite a cosmopolitan bunch as well and we get quite a few international catholics popping through (including many from spain) and i think the trend checks out -- even among the Spanish catbolics, the highly educated ones are more likely to be traditional