I think you are actually helping to prove my point. The global fertility rate has been dropping rapidly since 1970[1]. The poverty rate has also been dropping rapidly for just about as long[2]. Isn't is possible that the increase in living standards is being driven by the decrease in birth rate? That seems more likely than more people=less poverty.
> The reflex of many economists when thinking about the fertility rate is to point to income as the likely determinant. And sure enough, between countries and over time we see that higher incomes are associated with lower fertility. But good things come together – richer countries are also healthier and better educated – and so this correlation between high incomes and low fertility alone is surely not evidence that it is increasing income that is responsible for the decrease in fertility.
TLDR Major contributing factors are a woman's education (educated woman have fewer children, and delay having children until later in life), access to contraceptives, a reduction in infant mortality, with social norms and increases in income and quality of life to a lesser extent.
Sidenote: Thanks to Our World In Data (YC 19 NP) for performing and hosting top notch datasets and the analysis of such.
It looks like as the population is growing, we are becoming less poor, not more, which dismantles the rest of your argument.