I don't think it's special magic blood or genetic or anything else. This concept is far from the mind of the authors of scripture. In fact I'd go as far as to say that the idea of magic blood is exactly the error being made by people who think of modern Jews as uniquely and especially God's people.
And, yeah - I'd point exactly to what you said. A mixed multitude went out of Egypt. A suspiciously high number of those in the Torah who left Egypt had non-Hebrew names, even Egyptian names. Caleb was a Kenizzite, but became an elder of Judah. So what made one an Israelite was not who your parents were, but what life you lived and who you worshipped. In other words, faithfulness to the God of Israel, as demonstrated by following His commands.
But, I disagree that those Israelites who were scattered into the gentiles maintained this faithfulness. In fact I would argue the opposite - they had been unfaithful, and because of that unfaithfulness were scattered, and their identity as a people was wholly lost. They became gentiles, truly - not by genetics, but by nomos or Torah or pattern and way of living, including worship. And this is why when they come back, they do not come back as Israelites per se - because if they did St Paul's whole premise is wrong, and those who say they must follow the Torah as Israelites would be correct. They come back as faithful people of their own nations. So the NT is working out the idea - what does it mean to be a faithful Greek Christian? How does that look? St Paul is teaching this, and it itself is is a fulfillment of prophecy (Isaiah 66, Ezekiel 34, Amos 9, Micah 4 etc). Their way of living, the nomos of being Greek, is redeemed and becomes something new. They don't become Judaeans, or Northern Israelites, but Greek Christians, Greek Israelites, as it were. And so all twelve tribes are reconstituted in the prophecy of the dry bones, the promises and commands of Deuteronomy of the people of God and nation of priests are applied to gentile Christians by the Apostles, and the faithful of the tribes are seen in the Apocalypse of John (except Dan and Ephraim, which is another story..).
This is exactly the close reading of the Torah that the council of Jerusalem takes, for what its worth. They apply the Torah strictly as written - to the sons of Israel are commands, and to foreigners dwelling among them there are commands. They apply all of the commands which apply to foreigners strictly to the gentile Christians, upholding the law strictly.
And you are correct to see that this is a familial structure. We gloss over the use of "brethren" in the NT scriptures but this is applying the social construct, and reciprocal bonds of obligation, to Christians as a family. A household with overseeing slave stewards (the episcopos) and a patriarchal head in Christ. In the end, the beautiful picture of the restored house of God is All Israel, reconstituted. Thus, the problems of death, sin, idolatry, demonic domination, and separation from God presented in the Torah are solved through the movement of God toward the creation and Mankind that He loved. It starts with a single family, which becomes a completely new nation after the others were lost. That nation, despite its own failures and faithlessness, becomes the vehicle for the union of the Divine with Man's nature, elevating Man's nature by joining it to that of God in the Incarnation. At the Cross and Resurrection death is defeated. And, through the Ascension and giving of the Holy Spirit, the cleansing of the camp and that single tribe we see in the Day of Atonement is expanded, once for all, to cleanse and reclaim the whole world from the taint of sin, bringing all Mankind back from their enslavement to the demonic powers and opening the possibility of communion with God to all. So, one family, and the offspring of one man, ultimately become the means to reunite all Mankind back to one family again - All Israel is exactly this family.
This is why the very idea of magic blood in a small group of people being used to justify the death of so many is fundamentally twisted and at odds with the entire redemptive act, the main story as it were, of God in the scriptures.