Why did al-Bukhari and Muslim transmit hadiths from weak or abandoned narrators, even those who were labeled as liars by the imams of al-jarh wa al-ta'dil? And why is the methodology of selection not generalized to the compilers of the Four Sunan if this is the answer?

The narration by the two Sheikhs or one of them from a [jurh] (criticized narrator) has been explained and justified by the scholars. One can refer to the introduction of Al-Hafiz Ibn Hajar to his commentary on Sahih al-Bukhari (Hady al-Sari), where he detailed the names of those who were criticized and the answers to the objections. The inclusion of any narrator by the author of a Sahih [book] implies their trustworthiness in his view, and the consensus of the public on naming the two books "the Two Sahihs" is tantamount to a consensus on the trustworthiness of those mentioned therein.

However, if a narrator is included in follow-ups (mutaba'at), supporting narrations (shawahid), or hanging narrations (ta'aliq), their degrees vary. If a criticism is found regarding any of them, it is only accepted if the reason is explained and the flaw clarified. One can also refer to the words of Al-Hafiz Abu Amr ibn al-Salah in his book (Siyanat Sahih Muslim), where he clarified that Muslim narrated from some weak narrators for reasons, including: that the narrator might be considered weak by others but trustworthy by him, or that the narration is in supporting evidence and follow-ups, not in the main principles, or that the narrator's weakness occurred after the narration was taken from him, or to elevate his chain of narration. Ibn al-Salah concluded that ruling a person as meeting the criteria of Sahih merely because Muslim narrated from him is an error; rather, it depends on the manner and aspect of the narration.

It was noted that some of the names mentioned in the question are not among the narrators of the two Sheikhs or the Six Books, such as Nasih ibn al-Alaa. Others have no criticism regarding their trustworthiness or precision, such as Rawh ibn Ubada al-Basri.