If that happened, the population would plummet quickly, without any need for Draconian eugenics policies. People who expect their children to survive to adulthood have less of them; that isn't speculative, but a simple sociological fact.
People in Europe aren't different from people in Africa beyond the fact that one's personal space is more varied, less constrained by social necessity, permitting one to choose to avoid being encumbered with pregnancy out of necessity. People still choose sex in nearly as high frequency as do those who hve fewer other alternatives for pleasure.
True there is some evidence in places like Japan that intercourse rates are down significantly. that may be mostly because other options have become more attractive than the possibility of some need for pairing.
My objection is simply that people don't consciously make judgements on possibility of child reproductive capacity. It happens as a consequence of other factors coming between children having and doing something else.
All of this os off point about whether humans will exceed carrying capacity. In my studies, although a bit limited, I never encountered a species that,when having the capacity didn't exceed carrying capacity. I believe it's an ecological dictum.
You're a bit behind the times, and also know literally nothing about actual human demography. Europeans have less children per family unit every passing year.
Intercourse rates have been decoupled from reproductive rates since the introduction of effective, woman controlled, contraception in the 1960s.
The overpopulation crowd seem to be incapable of seeing this very clear and obvious fact, despite fifty years of supporting data that shows that humans who have a free choice in the matter, on average have fewer children than are necessary to sustain current population levels.
No coercive efforts are needed. Nothing needs to be done that has anything directly to do with population or reproduction at all:
We should provide an education to all children (particularly girls) who currently don't have access to an education.
We should provide healthcare - particularly preventative care such as vaccination and public health initiatives that provide clean water and effective sewerage - to those who do not have it.
We should encourage and enable people who want to earn money to do do so, and we should eliminate as far as possible the obstacles to learning and earning that still exist for the poorest people.
All three of these things should be done whether world population is a hundred billion, ten billion, or ten million - they are population independent acts of moral worth.
That we observe that all of these also tend to reduce (significantly) total fertility rates, and hence population, is a by-product. It need not (and indeed should not) be our focus. Because just as history shows that focus on education, health and wealth is good for people in many ways, so history ALSO shows that focus on population reduction is the source of a lot of really vile evil.
Population should never be highlighted as an important issue, for sound moral reasons. Population reduction is not a worthwhile enough goal as to justify the risks inherent in the widespread belief that population numbers are a problem in need of a solution.
We needed a solution in the mid-20th century. We found one. It's over. Living in the past isn't helping - we need to move on to the
unsolved problems, and stop pointlessly banging our heads against the now wide open door of 'overpopulation will be our doom'.
Sadly, humans tend to be about 30-50 years behind the times, all the time - people get agitated about the big issues in their 20s, but don't have the power to do something about it until their 50s, 60s or 70s.
Then they act - even if the problems are no longer there.