You'll probably find more than one version of this discussion - - but this is MY version:
In an evolutionary scenario, there is a moral difference between non-human primates and the creature God considers the First Human.
Naturally, how this definition is established is something that God decides, but it involves the first creature to be held "Morally Responsible" for his actions. What you would call "Original Sin" is the inherent weakness of flesh, or of non-divine creation.
The Genesis story makes it clear that even Adam's sinful nature would not prevent him from benefiting from eating of the Tree of Life. In fact, it is because even sinful humans can be immortal by means of the Tree of Life that God has to physically separate humans from the Tree... protected by an angel and the flaming sword.
What you want to call "Original Sin" might be better described as "Originally Sinful" ... for all come short of the glory of God. There must always be the First primate to be held morally responsible. And if the Baptists are correct about the useless nature of infant baptism, then even the very first moral hominid doesn't become MORAL until sometime before puberty...when the first moral hominid attains a moral maturity!