My bias here is that I tend to agree with the core thesis from the article: "without a state pension to lean on, children become the retirement plan" but with the even more unfortunate asterisk that even sufficient personal wealth is enough to discourage children, as seen by the correlations with birthrates in wealthier country. The "thing we can't say" here is that, bluntly, many people simply don't want kids vs. leisure.
Part of my reason for suspecting this is watching the struggles of various wealthy countries over the last 20 years to improve their birthrates with policies to make having children easier and giving incremental perks to those who do. My hypothesis for why none of these had a tremendous effect is that, simply, the magnitude of cost/benefit for a child is orders past the benefits currently being given.
Consider: For much of human history, your child was your retirement plan, a farmhand, a babysitter/daycare, and potentially a tool to align disparate families in marriage. This is not on the order of tens of thousands of dollars to have these perks in the modern world, they would cost hundreds of thousands into millions.
I totally believe there will be some who form collectives and opt-out, I would likely be one of them, but I have observed our world is not well set up for this sort of lifestyle, and I find most people are normative/not likely to do things off the beaten path.
This is a hard argument to make, I typically lean far more collectivist, and don't like the conclusions this leads me to in terms of what incentives will move the needle vs. not, but I'm struggling to see other paths that sufficiently impact the incentive structure.