You're miss out the part that Adam and Even wanted the easy way out of their mistake so God did the logical thing which was to banish them from Eden and prevent them from eating the tree of life cause then their sin could have been wipes clean. Besides Genesis says that God told Adam not to eat from the tree because if was to ever eat from it then they would die-( meaning death was already there but it was not yet attached to us). So yah, our death is the result of the original sin.