You wrote: "Is there any hope for a sharp reduction in the birth rate?
I believe that there is, given that such reductions in fertility follow on population trajectories that are already present in many developed countries. Historically, women married young and started having children immediately. Now the trend is for women to delay marriage and child-bearing while they pursue education and establish careers. Previously, children provided the primary system of old-age retirement and care. This is transitioning to a system of government-provided pensions and elder care. Small apartments in expensive cities favor few children, or none at all. Childlessness is becoming more socially acceptable than it used to be. These trends are likely to continue, and need to be supported by governments."
The primary system you are calling previous is unfortunately still considered the basis of their power as a group by sunnits, shiits, catholics, and other important religions. The Christan doctrine "be fertile and get more" is not far from the Mutterkreuz awarded by the nazis for mothers with at least four children. Unlimited growth of population is even anchored in less archaic principles of humanism. "Earth has room for all" is definitely a lie if women have ten ore more children as presently in the developing countries. That's why you may here expect much less support than those who are lacking conscience and cleverly evade the issue.
I appreciate not just your intention but also the scientific style of your essay and your answer to Joe.
Maybe, it is tactically imprudent to demand a reduction of world population by a factor of ten. I would prefer recognizing China's one-child-policy as a contribution that partially avoids otherwise even worse problems.
Moreover, I would prefer dealing not just with future overpopulation but also with not immediately related ongoing questions like unemployment, terror attacks, "religious" wars, slums, mounting risky waste, locally lacking water, etc. There is a lot to steer, and this needs to answer foundational ethical questions anew.