Ezra chapter 3 talks about Israel being returned from Babylon as "one man to Jerusalem" and the number of Israel was quite small. No doubt, they had been Purified. Thus, the question is: Who, exactly, is Israel throughout the Bible . . . and this question should be applied to the Sadducees, Pharisees, and teachers of the law as described in the New Testament. Who, exactly were these people?
So from the start, the entire nation was "Israel." When Paul talks about "true Israel" he is talking about the ideal, ie what the nation was called to live up to. And he then shows that it has little to do with racial qualities. If that was the criteria, then Ishmael, and Esau, being related to a patriarch would've been, by default, part of the chosen nation.
But God deliberately chosen a nation, descended from a line of *believers,* so as to later use faith as the criteria as to who in the end would make it. It starts with a nationality and a race, but ends with only those of faith being chosen.
It's the same thing with the Church today. Various groups always try to say that only their group is the true Church because only their group requires, doctrinally, that they all have faith. But the reality is, every group that starts with the highest qualifications always ends with some not meeting those standards.
So we just call Christians in various sects "the Church," and leave the judgment to God or to later on, when people really reveal who they're going to be in the long run.