The people who are having less kids in order to live longer, are those who are in the wealthier parts of the planet and are middle-class, and those who are very wealthy in the poorer parts of the planet, which is usually those here. The rest of the planet is having lots of kids. They just don't go on the internet much. They got more important things to do, like looking after their kids.
Actually, what you are talking about, is already going on.
Those who in the middle-class of Western countries, are having so few kids, that they have a negative population rate. Accordingly, there are less people than is required to replace them in the responsibilities that they fulfil. The new generation goes for the most socially impressive jobs, like in science, leaving the least socially impressive jobs, like cleaning the toilets, to someone else. However, Florence Nightingale showed that even in a war, 8 out of 9 deaths are caused by poor sanitation. So someone HAS to clean the toilets, no matter what, for us to survive. So those from poorer countries were encouraged to come to Western countries, to take up those jobs, as even the poor wage they got in those jobs, was substantially more than they were paid in their own countries for being a professional electrician. So you see, we can't really survive if we cut the birth rates for the entire planet as a whole.