This is going to be an extremely heterodox opinion. Please be charitable. As far as I know I'm the only person other than my wife with this opinion. I don't get this from anywhere. Then again, I don't read very much sociology so I don't know. I guess what I'm saying is I come by these views honestly.
My wife and I are pretty traditional Catholics. That is to say... we don't use birth control. We also got married pretty young. All things considered, I imagine we will have about 4 - 6 kids by the time she enters menopause. This informs my views.
Now to my points.
From a business perspective, people taking time off for their kids is disruptive to the business. Try as you may to incentivize businesses to get around this, the truth is that a worker that is present is going to be better than one who takes several weeks off every 2-3 years. Now, the typical response is that parents are more productive and dedicated as employees. That is perhaps true, but I doubt the parents of six children are any more productive or dedicated than the parents of one or two.
In order for a good fertility rate, you need most couples to have 2 or 3 kids, and some to have 4 or more kids, to account for those who do not marry or cannot have kids or cannot have enough to replace themselves.
If parents can potentially take up to a year off (as they can in Canada for example), which is incredibly disruptive to business, then companies are not going to want to hire them. This is not going to take the form of explicit 'no parents allowed'. But rather, all incentives ultimately 'leak' -- incentives are a leaky abstraction.
At some point the additional loyalty and hard work one gets from parents is going to be balanced off by the risk that you may have to give them one year off. Companies do not need to explicitly fire parents for this to shake out in the market. Parents are going to internally police themselves as well. Due to how much of a benefit being able to take a full year off is, parents are going to succumb to the natural social pressure of not demanding too much of society. If every kid is an excuse to take a year off, it's not long before parents with lots of kids are seen as welfare queens. We already see this in places like France, where President Macron has made remarks about how ambitious women don't have lots of children.
This also leads to an incentive to not have kids because those that put in that extra year that a parent may take off accelerate their promotions at a faster speed. Thus, kids grow up seeing that the childless are the most successful and then want to emulate that path.
I feel this pressure currently. With the birth of a new baby pending, and the likely addition of another child a few years after, if I were to take 12 months off as some european countries allow, I'd soon develop the reputation of that guy who takes years off at a time. Work one year, then leave the other. Already, it's common in the quiet recesses of the break room for child-free colleagues to complain about having to cover for the mom/dad having another kid. This sort of resentment will never go unheeded; the effect is just subtle. Ultimately, this will have the effect of making parents feel guilty, thus making them reconsider having a kid. I already feel guilty about taking time off for my next baby.
So, I believe these three effects -- the self-policing of parents who don't want to be seen as leeches, the natural advancement of the childless/those with fewer children over those with more, and the subliminal incentive to hire workers who are not going to take long, disruptive breaks every few years -- leads to a society in which only having some X number of children is really acceptable. In contrast, in American society, there is less incentive to be nosy in people's childbearing lives. Whether someone has six or ten, since society provides comparatively less to the kids, as long as the parents can provide for them, many people see childbearing as a personal decision. In a more collectivist society, where government does provide a lot of benefits, then having 'too many' children can become a source of social shame much in the way that the title 'welfare queen' carries a stigma of being on government benefits.
I don't believe such constraints can be overcome by more government regulation. Or if they can, I haven't seen anywhere that does so successfully. The effects are natural. Imagine if a 20 year old woman got a nice job in a Canadian company. Then at age 22, after two years of exemplary service, got married and had a baby every year until 40. This is a bit extreme, but certainly within possibility if she wanted it. Now she requests one year off each time. That woman is 'entitled' to twenty years off. Don't you think she'll face lower career prospects, less advancement, and less employment? Now, imagine she just takes every other year off or every fourth year off, etc? It seems to me these negative effects scale linearly and they do so simply because each child represents time taken out of the workforce, while your childless / less fruitful colleagues are advancing freely.
Back to my personal observations. I have a few acquaintances who are also having big families. guess what? Their paternity/maternity leaves are much shorter. They realize that if they took the full benefit, they will have less career advancement, less prestige, and reduce their chances of employment. Thus, they take less. In America, where it's normal to take a short paternity break (or maternity of only a few months), this carries little stigma, and they're just seen as hard-working. In places where most parents take many many months off, then these parents would be seen as bad parents.
Kind of a brain dump, sorry.