Or the question for the Catholics men and women especially and all Christians in general might be "will you allow God to be God?" You know that artificial birth control is "no births and no control." How about a call to be responsible? The Creighton method of Natural Family Planning has a proven "track record" equal to the pill (and without the awful side effects).
You know, when people can no longer replace themselves, the human race dies. And before that their standard of living gets less and less simply because there aren't enough people to maintain even the industrial revolution. Future generations will be tending their crops and hunting for food.
You're forgetting a few key factors here. One factor is increasing automation. As technology expands, there are less human beings needed to operate a factory of any given size, so much is done with computers and robots. Farming has also begun to see situations where technology has made things less man-power intensive.
Secondly, population growth combined with global climate change are likely to, barring significant technological innovation (Which I admit is possible), going to be significant stress on natural resources. We're already seeing rising food and fuel prices as the result of drought in the American midwest (the bread basket of the world). What happens when the American midwest is a desert? Scientists are also saying that changes in the climate may eventually make the ocean ecology change in ways that lead to the mass extinction of salt water fish, a major food source in some areas, with jelly fish taking over that ecosystem, which again hits the world supply, unless we learn to eat jellyfish (Which is possible, I suppose, but I think they'd need to be processed to remove the poison). Droughts are going to worsen and deserts are going to widen in already hard up places like much of Africa. It might be easier from a pragmatic perspective if we were able to slow population growth and have fewer mouths to feed and water in the world than we would otherwise (Not saying that there are moral ways to make that happen, or that we should make it happen, just speaking purely pragmatically and hypothetically).
Finally, just as an aside, even modern more scientifically accurate versions of the rhythm method is not as effective as the pill, and even where they do work to some degree, they require a lot of self-discipline that a lot of people don't have in practice.
Keep in mind, none of that is an attack on Catholic teaching on birth control. Who knows what God has planned in terms of the future of the world? All I'm saying is that world population growth, and alternatives to birth control, pose challenges for the human race as a whole (especially in light of global climate change) as time marches on, barring unforeseen changes, and for individual couples right now in the present. Of course, technology could help mitigate the former. And expanding into space could help with both in the much longer term. And no one ever said life was going to be easy.