It's a biological urge for many, which makes sense from an evolutionary standpoint. Sure some don't have it, and some control it, but that's clearly not the majority of our species otherwise we wouldn't be so ubiquitous.
I can't say strongly that you're wrong, but I can say your thesis is most definitely unproven, and there is good counterveiling evidence.
Clearly there has and remains enough historical reproduction that humans are pretty much ubiquitous and have affected pretty much the entire surface of the planet. No question. But the why is unknown.
Yes there are of course people who feel a need to duplicate their genes (though to my knowledge I've literally only met one person in my 50+ years who said that this was their desire). However different people have many different reasons to intentionally reproduce and demonstrably, when given the ability, a dominant number opt to limit it (this fact is the launching point of the original article).
The clearest example is economic: birth rate and income are pretty highly negatively correlated (the US is an outlier but when you double click and look at the income data at higher resolution than national, you see the same thing). I could go into examples of why this might be the case but that could lead to an unnecessary flame war.