The problem is you said that nobody is received as a Jew (ethnic) without being circumcised. This is not true (hence "a problem).
Jewish law (Halakha) considers uncircumcized Jews as fully Jewish.
There are exceptions in some Reform interpretations, but even among Reform Jews circumcision is required except in some situations.
You're just jumping across traditional and novel interpretations as if there's some consistency with each other or even with Moses, and it just isn't so.
But since you want to talk about Israel™ as if it's practices have any connection with the Israel of the Bible, or even Israel as defined by Judaism, let's talk about Israel.
I've had a book on my shelves for a number of years,
A Rabbi Talks with Jesus, by Jacob Neusner, a Reform rabbi. He was considered by many the foremost authority on Judaism in the first centuries of Christianity. His purpose was to explain why, if he were a Jew of the first Century, he would have rejected Jesus.
He spoke of the identity of Israel. There is an 'eternal Israel', by which he meant all Jews from Sinai forward, and of an 'Israel of the here and now,' by which he meant Jews practicing in a community. The Zionist state, is nowhere in sight. One's ethnicity is nowhere in sight. One's Jewishness is based on His connection to the law, not to the land, nor to his ethnicity. There is no ethnic Jew or nationalist Jew. Those are Zionist constructions.
Bibically, those born to Jews are Jews because of the Torah. Infants were circumcised because they were born to people in covenant with God, not because of their genealogies. So they were born Jews. Others became Jews by the observance of a carnal commandment. That's what it meant to be a Jew by nature, Gal. 2:15.
Back to the identity of Israel. The real Israel. Under the old covenant, it was the nation that entered into covenant with God by receiving the law. That old covenant has now been canceled. To try to resurrect it or to assert some aspect of it as still binding is antichrist. Israel is still the nation that has entered into covenant with God, but through the blood of Jesus, and not through the letter of the law. Those under the old covenant were Israel in childhood. Those who are in Christ, is the same Israel, but in adulthood. There is no other Israel. As some have already pointed out, he is not a Jew which is one outwardly, that is, a Jew by nature. But he is a Jew which is one inwardly, meaning by the Spirit, and circumcision is of the heart.
Biblically, that is the identity of Israel.
Jacob Neusner would have rejected Jesus, because, according to him, Jesus preached a different Israel than the Israel the Jews saw in the law, and a different kingdom, and a different temple...and Neusner was right.
Yakkov Shapiro, an Orthodox rabbi, In his book, The Empty Wagon, which now has a place on my shelves beside Neusner's book, begins his 1300 page tome with the question, What are the Jews? He said:
The Jewish religion is the only type of Jewishness that exists, and becoming more religious is the only Jewish goal.
and
If somehow it would be possible to take back the Torah from the Jewish people, there would then be no more Jews in the world.
It really doesn't matter what Israel™ says and does, and even less what that genocidal Netanyahu thinks and does or what he puts forth as his ancestry. It's probably a lie anyway. It has nothing whatever to do with the traditional identity of the Jews, and even less with the Jews of the Bible.
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