The impression I first got from this sentence was that you were wondering why animals die, since they don’t have original sin. The rest of your post doesn’t suggest that that’s your question, but just in case, it is my understanding that “mortal” is the natural state for material organisms. In a natural world before the Fall, and without divine intervention, we should expect both humans and animals to die. Only with God providing immortality by a miracle could Adam and Eve have lived forever. Scripture indicates this by saying Adam and Eve had to eat of the tree of life in order to live forever: Genesis 2:9, Genesis 3:22.
Therefore, it is my understanding that death is not absolutely a punishment for original sin. Only in the case of humans. I hope that helps.
I know that if God wanted to grant humanity with eternal life and stay on earth, he could easily do that, but wouldn’t that eventually overpopulate the earth if no one died?
Three options present themselves to my mind: one, they might have stopped having sexual relations after the earth was full. God had commanded mankind to “fill the earth and subdue it.” Once the mission was accomplished, perhaps they would have all been celibates.
Another possibility is that they would have developed space station technology and gradually populated the rest of the universe. That could buy them something on the order of billions of years before they would need to become celibates, unless technology can be used to make new universes.
A third possibility is that God would use His almighty power to create additional living space so that mankind would never run out of room or resources.
Also, aren’t we supposed to want to be with God, and not stay on earth?
Scripture indicates that God walked with men on the earth: Genesis 3:8. It is my understanding that, in Catholic teaching, mankind will continue to live an everlasting physical life on a renewed earth after Judgment Day: CCC 1042-1050. Because of this, I don’t think there is a dichotomy between staying on earth and being with God. God will walk with men again.
I’ve come up with a tentative solution, which makes sense to me at least, that would explain this. In their unfallen state, humans might have simply chosen to die and then be assumed into heaven.
I think I see a problem with this: in Catholic teaching, human nature is incomplete without the body. The Catechism says that the body and soul form a single nature, that the body shares in the dignity of the image of God, and that therefore, man may not despise his body. (CCC 364-365) In light of that, I can’t see what sense it would make for mankind to definitively stop their bodily existence. They would be giving up half their being and making themselves incomplete persons.
For all these reasons, I think it is safe to say that, if mankind had not fallen from grace, we would still continue to live a bodily existence. I hope that helps.