Abrahamic covenant and Brit Milah - how is the covenant passed down?

Question

Why is it so that if a Jew has a male child with a non-Jewish woman the child is not circumcised, even though, like Ishmail, the child is under Abrahamic (but not Mosaic) covenant, since the father is of "seed of Abraham"? (I assume Abrahamic covenant status follows the father, since it descends from Abraham to Ishmail and Isaac, and from Isaac to Jacob, from Jacob to all the other Jews and their offspring).

Also, such a child is not Jewish (not under the Mosaic covenant), but Genesis 15:18–21 promises all the descendants of Abraham specific land and to curse those, who curse them. This is different than for Bnei Noah where no such promises are made, so wouldn't that imply a sign of the covenant should be made for all the descendants of Isaac (Ishmail did not inherit the covenant:

וַיֹּאמֶר אֱלֹהִים, אֲבָל שָׂרָה אִשְׁתְּךָ יֹלֶדֶת לְךָ בֵּן, וְקָרָאתָ אֶת-שְׁמוֹ, יִצְחָק; וַהֲקִמֹתִי אֶת-בְּרִיתִי אִתּוֹ לִבְרִית עוֹלָם, לְזַרְעוֹ אַחֲרָיו
וּלְיִשְׁמָעֵאל, שְׁמַעְתִּיךָ--הִנֵּה בֵּרַכְתִּי אֹתוֹ וְהִפְרֵיתִי אֹתוֹ וְהִרְבֵּיתִי אֹתוֹ, בִּמְאֹד מְאֹד: שְׁנֵים-עָשָׂר נְשִׂיאִם יוֹלִיד, וּנְתַתִּיו לְגוֹי גָּדוֹל

)?

Answer 1

Because the lineage goes through the mother...kind of.

The primary source is a Mishnah in Tractate Kiddushin:

מתני׳ כל מקום שיש קידושין ואין עבירה הולד הולך אחר הזכר ואיזה זו זו כהנת לויה וישראלית שנשאו לכהן לוי וישראל וכל מקום שיש קידושין ויש עבירה הולד הולך אחר הפגום ואיזה זו זו אלמנה לכהן גדול גרושה וחלוצה לכהן הדיוט ממזרת ונתינה לישראל בת ישראל לממזר ולנתין וכל מי שאין לה עליו קידושין אבל יש לה על אחרים קידושין הולד ממזר ואיזה זה זה הבא על אחת מכל העריות שבתורה וכל מי שאין לה לא עליו ולא על אחרים קידושין הולד כמותה ואיזה זה זה ולד שפחה ונכרית

The Mishnah discusses four types of unions:

1. If the marriage is binding according to Jewish law, and there is no sin in the union, the child’s status follows the father. Therefore, if two ordinary Jews marry each other, the child takes the father’s status. Ex: if a Kohen marries the daughter of a Yisrael, the child is a Kohen.
   2. If the marriage is binding according to Jewish law, but there is a sin in the union, the child’s status follows the one whose status causes the sin. Ex: if a Mamzer marries an ordinary Jew, the child is a Mamzer, regardless of whether it’s the father or the mother who’s a Mamzer.
   3. If the marriage is not binding according to Jewish law, but it would be binding if each spouse married someone else, the child is a Mamzer. Ex: if a man married his sister, the child is a Mamzer, because each one can marry other ordinary Jews legally.

The one relevant to the question is the fourth category: where marriage is not binding according to Jewish law no matter which Jew one of the spouses marries. In such a case, the child’s status follows the mother, not the father. One example brought in the Mishnah is if a non-Jewish woman has a union a Jewish man.

As such, the child is fully Jewish if born from any of the first three types of unions, but he would potentially have a stigma on his lineage depending on the case. In the fourth type of union, though, the child is not Jewish since his mother isn’t Jewish. If he’s not Jewish, there’s no commandment to circumcise him!