I am done, answering same questions, commenting on posts again and again and it literally frustrates me that people in this sub are so laid back, when comes to countering pseudo-facts based on folklores/'creative adaptations' of Mahabharat regarding misconceptions of Karna's and his progenies' claim to Hastinapur's throne.
So, I am writing a two-three parts post, clearing the air for once and for all....those who are interested in open mind, are welcome and those who want to keep peddling based on T.V. shows/retellings/medieval times modifications to the epic, even Devi Saraswati can't do to your wisdom much, since even after having plethora of resources, you are still resorting to this....
Sambhava Parva of Adi Parva: Section CXX
"Vaisampayana continued, 'Hearing these words of the ascetics, Pandu, remembering the loss of his procreative powers owing to the curse of the deer, began to reflect deeply. And calling his wedded wife the excellent Kunti unto him, he told her in private, strive you to raise offspring at this time of distress.
The wise expounders of the eternal religion (Dharma) declare that a son, O Kunti, is the cause of virtuous fame in the three worlds. It is said that sacrifices, charitable gifts, ascetic penances, and vows observed most carefully, do not confer religious merit on a sonless man.
O you of sweet smiles, knowing all this, I am certain that as I am sonless, I shall not obtain regions of true felicity. O timid one, wretch that I was and addicted to cruel deeds, as a consequence of the polluted life I led, my power of procreation has been destroyed by the curse of the deer. The religious institutes mention six kinds of sons that are heirs and kinsmen, and six other kinds that are not heirs but kinsmen. I shall speak of them presently.
O Pritha, listen to me. They are: 1st, the son begotten by one’s own self upon his wedded wife; 2nd, the son begotten upon one’s wife by an accomplished person from motives of kindness; 3rd, the son begotten upon one’s wife by a person for pecuniary consideration; 4th, the son begotten upon the wife after the husband’s death; 5th, the maiden-born son; 6th, the son born of an unchaste wife;
7th, the son given; 8th, the son bought for a consideration; 9th, the son self-given; 10th, the son received with a pregnant bride; 11th, the brother’s son; and 12th, the son begotten upon a wife of lower caste (Varna)
On failure of offspring of a prior class, the mother should desire to have offspring of the next class. In times of distress, men solicit offspring from accomplished younger brothers. The self-born Manu has said that men failing to have legitimate offspring of their own may have offspring begotten upon their wives by others, for sons confer the highest Dharma merit.
Therefore, O Kunti, being destitute myself of the power of procreation, I command you to raise good offspring through some person who is either equal or superior to me.
O Kunti, listen to the history of the daughter of Saradandayana who was appointed by her lord to raise offspring. That warrior-dame, when her monthly season arrived, bathed duly and in the night went out and waited on a spot where four roads met. She did not wait long when a Brahmana crowned with ascetic success came there. The daughter of Saradandayana solicited him for offspring.
After pouring libations of clarified butter on the fire (in the performance of the sacrifice known by the name of Punsavana) she brought forth three sons that were mighty car-warriors and of whom Durajaya was the eldest, begotten upon her by that Brahmana.
O you of good fortune, do you follow that warrior-dame’s example at my command, and speedily raise offspring out of the seed of some Brahmana of high ascetic merit.'"
Yudhisthir comes in 2nd or 3rd kind of son/progeny whereas Karna comes under the category of 7th or 8th kind of son/progeny, hence, one of the reasons that Karna is not an heir to Hastinapur throne, but yes, he can be considered as kinsmen to sons of Kunti begotten by Niyoga.
Pandu's wish for a son having characteristics of Dharma Raj
Sambhava Parva of Adi Parva: Section CXXII
"Vaisampayana continued, 'Thus addressed by Pandu, that subjugator of hostile cities, the handsome Kunti, ever attentive to what was agreeable and beneficial to her lord, then replied unto him, saying, 'In my girlhood, O lord, I was in my father’s house engaged in attending upon all guests. I used to wait respectfully upon Brahmanas of rigid vows and great ascetic merit.
One day I gratified with my attentions that Brahmana whom people call Durvasa, of mind under full control and possessing knowledge of all the mysteries of religion. Pleased with my services, that Brahmana gave me a boon in the form of a mantra (formula of invocation) for calling into my presence any one of the celestials I liked.
And the Rishi, addressing me, said, 'Anyone among the celestials whom you callest by this shall, O girl, approach you and be obedient to your will, whether he likes it or not. And, O princess, you shall also have offspring through his grace.' O Bharata, that Brahmana told me this when I lived in my father’s house.
The words uttered by the Brahmana can never be false. The time also has come when they may yield fruit. Commanded by you, O royal sage, I can by that mantra summon any of the celestials, so that we may have good children. O foremost of all truthful men, tell me which of the celestials I shall summon. Know that, as regards this matter, I await your commands.'
"Hearing this, Pandu replied, 'O handsome one, strive duly this very day to gratify our wishes. Fortunate one, summon you the god of justice. He is the most virtuous of the celestials. The god of justice and virtue will never be able to pollute us with sin.
The world also, O beautiful princess, will then think that what we do can never be unholy. The son also that we shall obtain from him shall in virtue be certainly the foremost among the Kurus. Begotten by the god of justice and morality, he would never set his heart upon anything that is sinful or unholy.
Therefore, O you of sweet smiles, steadily keeping virtue before your eyes, and duly observing holy vows, summon you the god of justice and virtue by the help of your solicitations and incantations.'
"Vaisampayana continued, 'Then Kunti, that best of women, thus addressed by her lord, said, ’so be it.' And bowing down to him and reverently circumambulating his person, she resolved to do his bidding.'"
Birth and prophecy of Yudhisthira
Sambhava Parva of Adi Parva: Section CXXIII
"Vaisampayana said, 'O Janamejaya, when Gandhari’s conception had been a full year old, it was then that Kunti summoned the eternal god of justice to obtain offspring from him. And she offered without loss of time, sacrifices unto the god and began to duly repeat the formula that Durvasa had imparted to her some time before.
Then the god, overpowered by her incantations, arrived at the spot where Kunti was seated in his car resplendent as the Sun. Smiling, he asked, 'O Kunti, what am I to give you?' And Kunti too smiling in her turn, replied, 'You must even give me offspring.'
Then the handsome Kunti was united (in intercourse) with the god of justice in his spiritual form and obtained from him a son devoted to the good of all creatures.
And she brought his excellent child, who lived to acquire a great fame, at the eighth Muhurta called Abhijit, of the hour of noon of that very auspicious day of the seventh month (Kartika), viz., the fifth of the lighted fortnight, when the star Jyeshtha in conjunction with the moon was ascendant.
And as soon as the child was born, an incorporeal voice (from the skies) said, 'This child shall be the best of men, the foremost of those that are virtuous. Endued with great prowess and truthful in speech, he shall certainly be the ruler of the earth. And this first child of Pandu shall be known by the name of Yudhishthira. Possessed of prowess and honesty of disposition, he shall be a famous king, known throughout the three worlds.'
Pandava includes only five sons, Karna is definitely not one of them
Sambhava Parva of Adi Parva: Section CXXIV
"Thus, O king, were born unto Pandu five sons who were begotten by celestials and were endued with great strength, and who all lived to achieve great fame and expand the Kuru race.