Should good Buddhists encourage the extinction of all animal species?
I think you have misunderstood annihilism (no rebirth after life) as a teaching of Buddhism. You can’t end one’s suffering by ending his life, because Rebirth itself is the suffering (life itself may not be if you are wealthy and you are enjoying your life — you won’t view it as suffering, right?). In addition, physical suffering in life are karmic effect due to past unwholesome actions. If you end your life intentionally, the karmic effect would continue into your next life — that is, you continue to suffer (physically) in your next life. Extinction of any species is useless too since one can rebirth in any other species (human need not rebirth as human, and animals need not rebirth as animals). The right way to end suffering is to eliminate all afflictions by following the eightfold noble path.
