In fact, where is it not true?
No it isn't.
>In fact, where is it not true?
Everywhere? Nothing in your link contradicts what I said already. You can only make the correlation if you cherry pick start and end dates, and cherry pick western nations or nations that had coincidental birth rate declines like China. If the correlation is real and not spurious, then why doesn't it hold now? Why didn't it hold before the 1930s? Why has it never existed in Africa?
The inverse relationship is obvious in South Korea:
https://tradingeconomics.com/south-korea/gdp-per-capita
https://fred.stlouisfed.org/series/SPDYNTFRTINKOR
Before the 1930s, every country was "developing" my modern definitions. (US real GDP/capita was ~$9k in 1929 just before the recession). Also, other factors are necessary for this to happen (e.g. women empowerment)
No it isn't. That is based on a single observation, increasing standards of living in the 20th century in western nations. Living standards in western nations were increasing before that and birth rates were not declining. Living standards stopped increasing decades ago, and birth rates have not gone back up. So the evidence that this is even true in western nations is quite poor, but there is no evidence at all that it holds true in other cultures.
>We can't just exploit the world and then blame those who live in systems that offer no alternative.
It isn't about blame, it is about solving the problem. You can not give 5 billion Africans a western standard of living, that is not a physical possibility. So we need to prevent the 5 billion Africans from happening while there's still only 1 billion Africans. We're not trying to prevent 5 billion Europeans from happening because the European population is falling, not rising.