My perspective is that the OT must be read in light of the NT, as this reveals more of the big picture God intends for us to see. Originally in Genesis, the curse of death entered the world because of sin. I believe physical death is a picture of a more serious, death which is the separation between the person and God, the life-giving Creator. When God called the nation of Israel as a chosen people, gave them laws and the promised physical land, this was also a picture I believe which reveals the bigger promise of an eternal spiritual land to all who believe by faith, as Abraham did, and are therefore chosen to enter this eternal promised land. I think Paul had this overall larger picture in mind when he quoted from Deuteronomy in Galatians 3:13.
If a man has committed a sin deserving of death, and he is put to death, and you hang him on a tree, 23 his body shall not remain overnight on the tree, but you shall surely bury him that day, so that you do not defile the land which the Lord your God is giving you as an inheritance; for he who is hanged is accursed of God. Deut. 21:23
The passage in Deuteronomy, which Paul quoted, refers to the penalty being hanging on tree for the one who sins deserving of death. In Romans it is stated that for...all have sinned and fall short of the glory of God (3:23) and the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord (6:23). This goes along with the Gen. account that all humanity is under the curse of death, physical and spiritual, because of sin.
Paul realized and was teaching that all were under the curse of death because all were sinful and failed to keep the law. Yet, Christ kept the law and bore the sins of all on the cross (tree) so that all may have eternal life instead of death. The passage (in Deut.) also reveals that God did not want the physical land which was the inheritance defiled by sin and death, so the eternal heavenly land will have no sin, nor death.