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On the one side of the family, we have the union between Tamar and Yehudah, which, while technically a fulfillment of Yibum, Yehudah thought she was a harlot. From this union came Peretz (Bereishis 38:15-29), the forebear of David HaMelech (Rus 4:18-22).

On the other side of the family, we have Rus, a descendant of Moav. He was born from the union of Lot with his eldest daughter (Bereishis 19:36-37).

Continuing down the line, David himself married Bas Sheva, after seeing her beauty and sending her husband Uziah to the front lines of the war to be killed in battle such that she would be single (Shmuel 2:11:2-27). From this union came Shlomo HaMelech (ibid. 12:24).

David’s lineage was called into question by Doeg HaEdomi, until it was confirmed that a female Moavite convert is not prohibited (Yevamos 76b-77a).

David himself was thought to have been born from the union between Yishai and his maidservant, rather than his actual wife (Me’am Loez to Shmuel 1:16:11; cf. here).

Shlomo himself married Ne’amah, a descendant of Ben-Ami, the son of Lot with his other daughter (Bereishis 19:36,38). Ne’amah went on to be the mother of Rechavam (Melachim 1:14:21).

All of these incidents were either technically permitted or at least done for noble causes, yet smack of something illicit. David married a single woman, after her husband was killed as rebelling against the king (Kiddushin 43a), yet was reprimanded for it (Shmuel 2 ch. 12). Yehudah, even if he didn’t know it, was fulfilling the mitzvah of Yibum, yet thought that he was being intimate with a harlot. And Lot’s daughters are praised for their actions, as they thought that the world had been destroyed and it was up to them to continue the human race, but at the same time, their actions are termed “sins” for Heaven’s sake (Horayos 10b-11a). While Rus was allowed to marry into the congregation, it wasn’t without controversy. And David’s mother wasn’t confirmed to even be permitted to his father until the truth came out that she wasn’t a maidservant.

Why must the Davidic dynasty come from such a background? Why couldn’t it be a normal family with normal marriages?

See the gemara in Yoma 22b.

This was orchestrated in order to ensure the longevity of his kingdom.

As Rabbi Yoḥanan said in the name of Rabbi Shimon ben Yehotzadak: One appoints a leader over the community only if he has a box full of creeping animals hanging behind him, i.e., he has something inappropriate in his ancestry that preceded him. Why is that? It is so that if he exhibits a haughty attitude toward the community, one can say to him: Turnand look behind you and be reminded of your humble roots.

Sefaria's explanation based on Rashi spells out:

This is why David’s kingdom lasted while Saul’s did not, as David descended from a family with problematic ancestry, namely Tamar (see Genesis, chapter 38) and Ruth the Moabite (see Ruth 4:18–22).

Good question.

(1) We each have two parents, four grandparents, eight great-grandparents, and the numbers keep growing exponentially in powers of two. So each one of us has A LOT of ancestors – thousands upon thousands. The chances are overwhelming that some of these people did not rank high on the scale of virtue. In fact, the Talmud tells us: Why was humanity derived from a single couple, Adam and Eve? So that no one should be able to tell his neighbor, "My ancestors were better than your ancestors". [Sanhedrin 37a] So, before you ask, “Why is God making the Messiah come from such dubious ancestry?”, ask yourself, “Does God really have a choice?”.

(2) The 19th century Hasidic master Rabbi Zadok HaKohen of Lublin writes, in his book Tzidkat Ha-Tzaddik:

“The redemption will emerge precisely from a place of lust and sin, by means of repentance (teshuvah). The Talmud says, in Sanhedrin 98a, “The son of David [i.e., the Messiah] will come only in a generation that is entirely wicked.” David is the archetype for the messianic soul because he showed how to make repentance into a life principle… And just that is the realization of ultimate fulfillment – that the root of evil will be transformed to good… At that time, the lowest will become the highest.”

(Actually, a direct check show that Sanhedrin 98a says: “Rabbi Yohanan also said: The son of David will come only in a generation that is entirely wicked or entirely righteous.”)

As the frumteens Rav wrote: (i'm paraphrasing from my memory of what he wrote) The satan is always poised to foil the ultimate purpose of Creation, which is the coming of Moshiach and the resulting acceptance of the Kingdom of Hashem upon all humanity. The satan will do anything to stop this ultimate victory of God's universal recognition and malchus. In order to fool the Satan, and so he wouldn't interfere with the coming of Moshiach, which is the ultimate good, and thus Satan's chief adversary, God made it so that the unions that would ultimately produce moshiach would appear to be improper, so that the Satan would never suspect that these unions were the forerunners of Moshiach, so he wouldn't interfere with them.

Because it's suppose to bring into question, rumination and idealization the very idea and notion of what it is to be human and that of course comes with sin and of course Death. No matter how much we try and sugar coat it, forms of harlotry and whoredom, indiscretion, mans failure to act on such and in cases act, females inability and ability to discriminate taking into consideration her carnal nature and dangers of such nature. Shepard's who see them self as failures in a society of serpents who preach them selves as successes accompanied by the ever seduced and tricked woman who suffers as such. Humans ability to convince them selves as society increases in complexity as well as less in nature that the world is to be one way while women are there to remind them the duty of the species as such while being blind to but a victim of their own folly. In God, man and woman are one. If you study the epic of Gilgamesh and the Odyssey, other creation myths, the idea of Atenism and its destruction by the Egyptians, cedar forest, Mesopotamian myths, Amarna letters and others you notice a pattern. A pattern in which emphasis marriage, the doom of woman on her own and male like wise, sex(duh I mean we are gonna die) the benefits of continued union; that and the formation of the idea of the self and civilization, God and death - man as civilized. Something that is to be seen as more human than immoral from the modern orthodox stance. Something so human that to say it outright would be akin to claiming to be a God of sorts, so lets just tell the little boys and troubled girls that it's a story and it has to do with God(s) and maybe it was told at a time when it as advice would have greatly helped a lost individual - an orphan perhaps or a woman caught in prostitution, a son to follow and lead a successful life or to not and fall to death and dare say a state of lost solo living (of which many times would find "it" out in such a state and to not reject it), which is certainly NOT Hu-man.I can go on.... Rabbi's did and continue to do a really good job of dissecting these tales of man - deciphering it, studying and indexing into a Bible of such high applicability and academic rigor and interpretation, building on that knowledge to the point of creating 3 religions and using what is built and solely as such to continue to advise man. Civilized man that is - as he is to be.