You are swapping correlation and causation to some degree. A country does not become industrialized by people starting to have kids at a later age. Rather, people start getting kids when their circumstances allow it: in industrialized countries, you rely less on children to provide for you when old, as there hopefully are social systems in place or you can save up on your own. Downside is, without social systems you also have to provide for yourself at old age, meaning people need to build up more savings before they feel ready for the financial burden a child is for around 20 years.
In developing countries, children often get little support above bare necessities and start contributing to the household income at a much earlier age, even before hitting their teens.
A country does not become industrialized by people starting to have kids at a later age.
There is a theory that supports this:
A core mechanism of unified growth theory is that accelerating technological progress induces mass education and, through interaction with child quantity-quality substitution, a decline in fertility.
Declines in fertility have been observed after a country has become industrialized. Not only did fertility decline, but the children people were having were generally ‘of higher quality’.
https://onlinelibrary.wiley.com/doi/full/10.3982/QE1751
The testable predictions of the theory and its underlying mechanisms have been confirmed in empirical and quantitative research in the past decade, and have inspired intensive exploration of the impact of historical and pre-historical forces on comparative economic development and the disparity in the wealth of nations.
This comes from Wiki, and this particular statement currently has 3 citations if anyone is interested: https://en.wikipedia.org/wiki/Unified_growth_theory
