Our Gemara on Amud Beis tells us how they inquired of Rabbi Yehoshua to rule over the dispute of Shammai and Hillel regarding the co-wife of a forbidden relation who falls to Yibum. He declares that he does not want to “stick his head between two great mountains”, i.e., between two great disputing opinions, between Beit Shammai and Beit Hillel, “out of fear lest these two mountains break my skull.”

Ben Yehoyada wonders why, specifically in this situation, Rabbi Yehoshua was so reluctant to offer a ruling. One answer that he gives is that since the ruling would have impact on future yichus, genealogy, he did not want to mix in. Since according to Beis Hillel the co-wife is forbidden to yibum and according to Beis Shammai, one may perform Yibum to the co-wife of a forbidden relation, the offspring from such unions could be permanently pagum (spiritually repugnant) to Beis Hillel for generations.

Simply put, Yichus is a touchy subject and Rabbi Yehoshua did not want to rule in such a way that disqualified the lineage of Shammai. 

While we are on the topic of lineage, it is worth noting the Jewish tradition’s ambivalence about this value. On the one hand, there is a long standing tradition, going back to antiquity, that values a person not just for his or her own achievements, but also for their ancestry. As with many values in Judaism, often there is a dialectic between two poles, in order to capture certain non-linear truths that need consideration to varying degrees, such as we saw in Psychology of the Daf, Yevamos 6.

Related to this, Abravanel (Numbers 4:21) comments on the order in which the Levite lineage is listed. On the one hand, the sons of Kehas were given the most honored duty in the Mishkan, that of tending to the holy vessels such as the Ark. However, in order not to subvert the hierarchy, Hashem asks Moshe to treat the sons of Gershom with equal reverence, using the language “גם הם Also them” to imply equivalence..

I will conclude with words from the Orchos Tzaddikim (27) about personal status versus lineage:

⁦"With three crowns was Israel crowned : the crown of the Torah, the crown of the Priesthood and the crown of Kingdom" (Aboth 4:17, Yoma 72b, Eccl. Rabbah 7:1). Aaron merited the crown of the Priesthood, as it is said, "And it shall be unto him, and to his seed after him, the covenant of an everlasting priesthood" (Num. 25:13). David merited the crown of Kingdom, as it is said, "His seed shall endure for ever, and his throne as the sun before Me" (Ps. 89:37). As for the crown of the Torah — it lies there waiting for all Israel, as it is said, "Moses commanded us a law, an inheritance of the congregation of Jacob" (Deut. 33:4) : anyone who wants the crown of Torah can come and take it. And should you say that the other crowns are greater than the crown of Torah...the Sages said, "A manzer who is a scholar takes precedence over a High Priest who is ignorant" (Horayoth 13a). As it is said, "She (wisdom) is more precious than rubies" (Prov. 3:15). The Hebrew word "peninah" means both "ruby" and "inside", and the verse can therefore be interpreted as meaning : More precious is the Torah and he who studies it than the High Priest who enters the very inside of the Temple — into the Holy of Holies.