Actually I'd argue that you can see this in action in birth rates right now in the USA and across most of the developed world. Having children is expensive. The standards for care and education keep getting higher. The available time for childcare is decreasing. It takes more and more education to be a fully participating adult member of society. That's a clear market signal to have less children and people are responding. The birthrate continues to fall and is below replacement in many countries, including the USA.
As a species, we appear to be regulating ourselves to prevent a catastrophic overshoot.
Also, in a world of 10 billion peoople, it's not like any given individual has much precious DNA to pass on.
I have one daughter and I'd love her to have a sibling, but for various reasons I don't think it's going to happen, which kind of makes me sad, but not because I won't pass down my "line".
I've noticed a strange phenomenon of people making statements like this and yet still have their own kids. I guess convincing potential competitors to stop having offspring would be a great way to ensure the success of your own genetic line.