Genetics were not what determined the parentage of a person. Like Dhritarashtra and Pandu, despite being Vyasa's sons, were not considered Brahmins. Vyasa did have sons by his own wife, and those were Brahmins. However, the sons he produced in Ambika and Ambalika were considered Vichitravirya's sons, since those 2 were Vichitravirya's wives. Similarly, Vidura too was treated as a son of Vichitravirya, since his birth was the equivalent of Ambika turning over a maid to her husband, and in that society, kings had sexual ownership of not just wives, but maids as well. However, since his mother was a shudra (not suta), he was not entitled to become king, but that didn't mean that he himself was a Shudra.
Also, even though he may have been Dharamraj incarnate, that didn't make him Yudhisthir's biological father any more than Rama was a father of Pradhyumna. Yudhisthir's biological father was Dharamraj, true, but not in Vidura's form. Besides, Kunti herself was a kshatrani - sister of Vasudev and married wife of Pandu, so no son of hers would have had any problems being king. However, Vidura and Yudhisthir did share a special bond due to the fact that one was Dharma reborn while the other was his son. Similarly, the Pandavas were not considered devas like, say, Indra's son Jayanta: they derived their rights from Pandu, the husband of their mom, rather than Indra, Yamaraj or Vayu, who actually produced them in Kunti
I am confused about this reply and the usage for it. I never said that the genetics would have had any effect of the parentage. I never said that a person has the caste of his biological parents
Dhritrashtra, Pandu and Vidur all three were considered Kshetraj santaan to Vichitraveer. Born by Niyog, none of them would have been considered Vyas Putra. But a Dasi Putra can not inherit. Manusmriti is pretty clear about it, they are given all the respect, support and nurturing that the other sons of a person (through his wife) gets, but Inheritance isn't for them. Shudra or not Vidur couldn't have have had inherited from Vichitraveer ( even if he was the biological son of Vichitraveer by a Dasi) why go far, even in the same family, in the next generation, Yuyutsu was elder to Duryodhan and Yudhishtir and was the son of Dhritrashtra, yet he was never considered for the throne because he was a Dasi Putra. That however doesn't mean that Vidur couldn't have wanted to become a king, he however knew it wasn't possible so never actively tried for it
Coming to the second part, we know that it's mentioned that Pandavas are the kids of gods, but if you remove divinity, this becomes impossible. Without divinity they were normal human born children born by Niyog of Kunti/Madri with humans. This is the reason, many historians do believe that Vidur (being considered the Dharmaraj incarnate) was actually the biological father of Yudhishtir. Also HearMeRoar said that it is also implied by Vyas that he was the son of Vidur. We are discussing that possibility here