David Moss, the President of the Association of Hebrew Catholics, has presented well, Father Elias Freidman's position on why he referred to us as Hebrew Catholics, rather than Jewish Catholics or Israelite Catholics. However, in his book Father Elias did not see the Old Covenant as irrevocable which included Law and Election but that only the Election was irrevocable. When he wrote his book this was still an open question until Pope John Paul II clearly stated that the Old Covenant was irrevocable and then put it in the Catechism.
Father Elias did not see Rabbinic Judaism as having a positive role in the economy of salvation which would later be taught by Popes John Paul II and Benedict XVI. Father Elias was only stating the position of many before the 1980's and I am sure he adjusted his position later in life to accord with the Catechism.
"The power of the community to mediate the "election factor" derives from the Election itself. The people of Israel, being the material object of the Election, each of its constituent communities is capable of mediating the "election factor"."
"To sum up, a community is Jewish in the sense of the term fixed by historical convention when it is ruled by rabbinical law; it is Israelite when it is in historical relation with the People of Israel prior to their dispersion from the Holy Land."
"The Christian is perhaps more aware of the catastrophe from which Rabbinism emerged than Chouraqui may imagine. Anyway, Chouraqui would surely agree that if what he says is true and God is in exile from the post-Christic Jewish People, the latter is automatically disqualified from playing a positive role in the economy of salvation. Consequently, we reject the affirmation of Gregory Baum according to which “Judaism continues to exercise a positive role in God’s plan of salvation”. What role did devolve on it we hope to consider later. If post-Christic Judaism were in any way valid, as Fisher pretends, the Jewish convert would be obliged to practice it even after his entry into the Church, which no one would be prepared to concede. The invalidation of Rabbinical Judaism should not be received as an offense. After all, Judaism invalidates Christianity."
"...The Marranos were crypto-Jews who practiced Catholicism in public and Judaism in secret. The majority ended up by being absorbed into the Spanish Church. A handful, especially on the island of Majorca, still keep up Jewish customs, even tending to intermarry. These habits are insufficient to regard them as Israelites. They are Spanish Catholics with judaizing tendencies..."
“...It results from a transcendental relation between the person and the divine will, mediated by the community of the elect. It is because one is born into the Elect People that one is born an Israelite. The ‘election factor’ is irrevocable for the person so born, since the gifts of God are without repentance. It is revocable for his descendants, not by an act of will, but where the descendants have ceased to belong to a community, mediator of the ‘election factor’.
I think it is pretty clear from the writings of Pope Benedict XVI that when the Church and the Catechism speak of the Old Covenant it is the Mosaic covenant as the core Covenant which includes the Abrahamic Covenant (which preceded) and the Davidic Covenant (which came after) the Mosaic Covenant. His comment about "the New Covenant as ratified at the Last Supper... was a prolongation of the Covenant of Sinai, which was not abrogated but renewed" makes that clear.
Cardinal Newman's understanding that with each development of doctrine and each Ecumenical Council of the Church there are elements of continuity and discontinuity can help in understanding Covenant theology. When the Covenant with Abraham was given to him and his seed (that was to become Israel) there was a continuity of the Covenant with Noah but also an element that changed to now include the role of the chosen nation or people from among the other nations.
In the same manner, with the Covenant given to Moses there was a certain continuity with the Abrahamic Covenant (including the Covenants made with the other Patriarchs who descended from him) but a new dimension was added which brought about changes in some outward observances. In the same way with the Davidic Covenant certain promises given in the Patriarchal Covenants were made clearer and clarified and while the Mosaic covenant continued it was in some ways transformed by the Davidic Covenant and new practices and structures occurred to include the role of the Davidic King which later was transformed due to historical circumstances into the role of the Davidic Nasi in the Sanhedrin. The seeds of this Davidic covenant were already found in the Adamic covenant in Genesis 3:15.
Thus the prophetic reflection on the Old Covenant by Jeremiah led him to enrich the tradition with an expectation of a coming future new Covenant and Malachi to a new sacrifice or universal oblation which culminated in the New Covenant made at the Last Supper. Thus, there would be elements of continuity and discontinuity in this regard for those who accepted the Messiah in his first coming. However, just as when God gave the Abrahamic covenant, the Covenant of Noah continued to apply to the Gentile nations- they were unknowingly spiritually enriched in a hidden manner by the coming of the Abrahamic Covenant to which they would be formally linked with their acceptance of the New Covenant in the times of mercy for the Gentiles. In a like manner, the Jews who did not accept the New Covenant did receive, in an unknowing manner, the graces of the redemption which they would fully embrace in the eschatological future with the Ingrafting. That the Jews, after the Temple was destroyed, adapted their observances to that historical reality does not mean they suddenly became something different, just as when the Davidic kings ceasing to reign and the move to the role of the Davidic Nasi didn't mean they were something different.
The Vatican under Pope John Paul II confirmed that Judaism over the last 2,000 years had a spiritual fruitfulness. We must remember that Father Elias wrote his "Jewish Identity" before the deeper discussions on Jews and Judaism in the light of Nostra Aetate and Vatican II ordered by Pope John Paul II in 1982 bore fruits as reported in the following Vatican document in 1985.
".... We must remember how much the balance of relations between Jews and Christians over two thousand years has been negative. We must remind ourselves how the permanence of Israel is accompanied by a continuous spiritual fecundity, in the rabbinical period, in the Middle Ages and in modern times, taking its start from a patrimony which we long shared, so much so that "the faith and religious life of the Jewish people as they are professed and practiced still today, can greatly help us to understand better certain aspects of the life of the Church" (John Paul II, 6 March 1982). Catechesis should on the other hand help in understanding the meaning for the Jews of the extermination during the years 1939-1945, and its consequences...." (From the 1985 -Notes on the Correct Way to Present the Jews and Judaism in Preaching and Catechesis in the Roman Catholic Church by the Vatican Commission for Religious Relations with the Jews)
Old Law and New Law: A Hebrew Catholic Perspective