We won’t stop breeding,
There was a movie “Zardoz” back in the 70s with Sean Connery. It was mostly awful, but it dealt with some of these issues.
“Er...don’t be too sure about that. I can imagine a world where an immortal elite will do anything to keep living on the kind of planet they want. Including ensuring that no one new is born once they feel the population is “ideal”.”
Look at what the population controllers are already doing. They’ve been doing this now for 50 years, and have been mostly successful.