PATERNAL FAMILY HERITAGE
במדבר פרק א, ב שְׂאוּ אֶת-רֹאשׁ כָּל-עֲדַת בְּנֵי-יִשְׂרָאֵל לְמִשְׁפְּחֹתָם לְבֵית אֲבֹתָם
מסכת בבא בתרא דף קט/ב משפחת אב קרויה משפחה משפחת אם אינה קרויה משפחה
Simply the males build the material aspects of civilization, and are thus called “builders” (בנים). As stated in Bereshis 5, 28, “a son was born to him that built the world”. And as the Ralbag explains a still-born baby is not called בן, since the baby did not have a chance to build the world.
Many practical differences deriving in one’s family heritage being from the father and not the mother. Maternal-only brothers and sisters do not inherit one another. Only paternal family members are responsible to redeem someone that was sold into slavery.
This concept can also assist in understanding the principle עמוני ולא עמונית מואבי ולא מואבית (this seasonal time of reading the book of Rus). The exclusion exists only for the males, since they are the main component of that civilization (that was banned from converting to the Jewish nation).
Also, the law recognizes greater responsibility to paternal family members in matters of charity and chalitzah. Alternatively, the Gemara (Kiddushin 68b) states that maternal-only brothers have greater love for one another than paternal-only brothers.
EVEN DOVID HAMELECH CAN NOT DO THE JOB OF THE LEVIIM
במדבר פרק א, נא וּבִנְסֹעַ הַמִּשְׁכָּן יוֹרִידוּ אֹתוֹ הַלְוִיִּם וּבַחֲנֹת הַמִּשְׁכָּן יָקִימוּ אֹתוֹ הַלְוִיִּם וְהַזָּר הַקָּרֵב יוּמָת
מסכת שבת דף לא/א
אמר ליה ההוא עובד כוכבים להלל מקרא זה על מי נאמר א"ל אפי' על דוד מלך ישראל
This Gemara is in the middle of the story of the non-Jew that came to Hillel and asked to be converted on the condition that he could become a Cohen Gadol. Hillel told him to study the entire Torah first. And when the non-Jew started to learn this verse, he approached Hillel and asked him to whom does this verse reference. And now Hillel told him even to Dovid HaMelech could not do the service of the Leviim (let alone the Cohanim). The non-Jew understood and then converted without the condition.
Why did Hillel mention the King? Reference is to another Gemara (Kerisus 6b) regarding the prohibition to benefit from anointing oil used for the Cohanim. In that Gemara a King is not included in the prohibition, since a King is not included in the category of an unauthorized person (זר). A King is also considered an anointed person. Nevertheless, in the matter of an unauthorized person for the service of the Leviim, a King is included in the prohibition.
Why did Hillel specify Dovid HaMelech? This alludes to the concept that even though converts are beloved to HaShem, and Dovid is descendent from Rus (probably the most famous convert), and Dovid was a King, nevertheless, this verse even applies to him. Further in Divrei HaYamim, one sees that Dovid desired to carry the Aron, he was specifically told that he could not since this was a job for the Leviim. How much more so, the prohibition to be a Cohen Gadol would apply to Dovid and thus, also, to this potential convert.