The extinction could be gradual, but I'm not a fan of that route since while a baby being born for a specific purpose and for the benefit of the remainder of humanity is assuredly better than one being born for the common reasons (or no reason) it is still selfish.
Since this plan could only ever work with global collaboration in the first place I believe that by combining the resources and knowledge of all countries we could build automated facilities for the final generation to live out their final days in relative comfort. If we started tomorrow we would have a solid 50 years or so to do it, and probably longer than that.
Even in a situation where something like that didn't happen and everyone was just left to find for themselves the amount of suffering would be large but magnitudes smaller than the amount our normal everyday way of living produces and the most important thing is that it would be a finite amount of suffering.
This isn't philosophy, this is fantasy.
You will never convince the human species to stop creating babies. There is no kernel of realism contained.