If you are going to equivocate with words from one message to the next it's going to be very difficult to dialog, as we have already discovered.
I realize this is uncomfortable for you because it strikes at the heart of your theology, but the fact remains that modern Israel does not fit the definition you gave for a "functioning Jewish state in Israel". In particular I note that they are not "Torah-observant".
Which brings us to the progress of the gospel. A Torah-observant Jew in the 1st century, just prior to the fuller initiation of the Kingdom with the coming of Jesus Christ, was a person obedient to the limited revelation found in the old covenant. Torah-observant meant keeping the festivals with their animal sacrifices, going to temple to worship God through the sacrifice of animals, keeping the lawful commands of the Jewish religious leaders. It meant keeping all the (limited) information God had given until that time. Many of these commandment were of a temporary nature and about to "fade away" (Heb. 8:13).
However, we no longer live in that world. Jesus and His apostles taught us that Torah-observant now means recognizing Jesus Christ as Messiah of Israel, King of kings and Lord of lords. It means trusting in His sacrifice, the blood shed once for all, and following His teachings, the true Torah of God. When believing Jews and gentiles came together in the Church, and observed all the apostles teachings (Acts 2:42), they were truly being Torah-observant.
Membership in Israel has always been by covenant not by "ethnicity", otherwise Ishmael and his descendants would be a member of "Israel". Under the old covenant the sign of the covenant was circumcision. Anyone, normally a Jew but also a gentile, could receive the sign of the covenant and become part of Israel and partake of the ordinances (cf. Exodus 12:44), whether they were physically descended from Abraham (ehtnically Jewish).
Once Jesus Christ came, they method of identification with Israel changed. It was no longer according to the bloody rituals of the old covenant, the cutting of the foreskin and sacrificing of animals. It was by the new covenant sign of identification, baptism, and the blood of the true Lamb of God who takes away the sins of the world. We no longer participate in the bloody sacrifice that pointed forward to Christ, but we have fellowship in Christ with fellow believers around the Lord's Supper of bread and wine. "Then He took the cup, and gave thanks, and gave it to them, saying, 'Drink from it, all of you. For this is My blood of the new covenant, which is shed for many for the remission of sins.'"
BTW, an adopted son, which we all are, has full rights and privileges of the father, including right to the "family name". If I as a gentile have been adopted into Israel, which you admit, then I can claim the right of the name Israel just as the natural sons. A gentile is not a second class member in the family of God. Otherwise your definition of adoption is stunted and hardly fits the biblical model.
Now I say that the heir, as long as he is a child, does not differ at all from a slave, though he is master of all, but is under guardians and stewards until the time appointed by the father. Even so we, when we were children, were in bondage under the elements of the world. But when the fullness of the time had come, God sent forth His Son, born of a woman, born under the law, to redeem those who were under the law, that we might receive the adoption as sons. And because you are sons, God has sent forth the Spirit of His Son into your hearts, crying out, "Abba, Father!" Therefore you are no longer a slave but a son, and if a son, then an heir of God through Christ. (Gal. 4)
Amen.