Paul obviously hadn't read the Tanakh.
It states clearly and at length that sin can't be inherited ─
Ezekiel 18
1 The word of the LORD came to me again:
2."What do you mean by repeating this proverb concerning the land of Israel, 'The fathers have eaten sour grapes, and the children's teeth are set on edge'?
3. As I live, says the Lord GOD, this proverb shall no more be used by you in Israel.
4. Behold, all souls are mine; the soul of the father as well as the soul of the son is mine: the soul that sins shall die. 5."If a man is righteous and does what is lawful and right--
6. if he does not eat upon the mountains or lift up his eyes to the idols of the house of Israel, does not defile his neighbor's wife or approach a woman in her time of impurity,
7. does not oppress any one, but restores to the debtor his pledge, commits no robbery, gives his bread to the hungry and covers the naked with a garment,
8. does not lend at interest or take any increase, withholds his hand from iniquity, executes true justice between man and man,
9. walks in my statutes, and is careful to observe my ordinances--he is righteous, he shall surely live, says the Lord GOD. 10 "If he begets a son who is a robber, a shedder of blood, 11. who does none of these duties, but eats upon the mountains, defiles his neighbor's wife, 12. oppresses the poor and needy, commits robbery, does not restore the pledge, lifts up his eyes to the idols, commits abomination,
13. lends at interest, and takes increase; shall he then live? He shall not live. He has done all these abominable things; he shall surely die; his blood shall be upon himself. 14. "But if this man begets a son who sees all the sins which his father has done, and fears, and does not do likewise,
15. who does not eat upon the mountains or lift up his eyes to the idols of the house of Israel, does not defile his neighbor's wife, 16. does not wrong any one, exacts no pledge, commits no robbery, but gives his bread to the hungry and covers the naked with a garment, 17. withholds his hand from iniquity, takes no interest or increase, observes my ordinances, and walks in my statutes; he shall not die for his father's iniquity; he shall surely live.
18. As for his father, because he practiced extortion, robbed his brother, and did what is not good among his people, behold, he shall die for his iniquity. 19. "Yet you say, 'Why should not the son suffer for the iniquity of the father?' When the son has done what is lawful and right, and has been careful to observe all my statutes, he shall surely live.
20. The soul that sins shall die. The son shall not suffer for the iniquity of the father, nor the father suffer for the iniquity of the son; the righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself. 21. "But if a wicked man turns away from all his sins which he has committed and keeps all my statutes and does what is lawful and right, he shall surely live; he shall not die. 22. None of the transgressions which he has committed shall be remembered against him; for the righteousness which he has done he shall live. 23. Have I any pleasure in the death of the wicked, says the Lord GOD, and not rather that he should turn from his way and live? [...]
30."Therefore I will judge you, O house of Israel, every one according to his ways, says the Lord GOD.
And if anyone would like to argue that the Garden story is somehow different, note that
the Garden story never mentions sin, original sin, the fall of man, disobedience, death entering the world, spiritual death, the need for a redeemer, or anything of that sort. Instead it states very clearly why God chucked Adam and Eve out of the Garden:
Genesis 3:22: Then the LORD God said, "Behold, the man has become like one of us, knowing good and evil; and now, lest he put forth his hand and take also of the tree of life, and eat, and live for ever"-- 23 therefore the LORD God sent him forth from the garden of Eden [...]