Originally anyone could convert to Judaism. Err, by originally, I mean for a few hundred years after Moses. At some point they got legalistic, and distorted the law/covenant. The original covenant is precisely the same as the current covenant, which is only honored by Christians - judge not, love your neighbor. The reason Judaism changed is also the same reason Christianity has undergone its own changes - power. Ever notice how baked into their (both) beliefs power is? Its always the same process. Purity tests : you're not one of us unless you believe/perform this extreme thing ; or some other invented legalism meant to exclude. In the Old Testament, this caused the Israelites to become idolaters over and over. Ezekiel, Isaiah, the exile - all because of idolatry. Idolatry is worshipping images. Law is an image. Materialism is equally an image. I've heard speculation that the genealogies aren't as ancient as they appear, that they were inserted for the purpose of deriving kingly legitimacy. Idk. I only listen to experts, I'm not one myself. I think it makes sense. Who was writing, and who was reading? Not regular people, they were illiterate. Anyways, I could've skipped all this and just said that matrilineal Judaism is a post-diaspora invention, but... I enjoy my verbosity.