This is how I understand the claim that the Apostolic Church, East and West, is messianic Judaism. I believe it is close to what Agrarian was saying.
If the Jews of the 1st century, en masse, had accepted Christ, then they and their heirs to this day would have developed a church that would have been in essentials identical to the Catholic or Orthodox Churches. The differences with what we actually have today and what they would have developed would have been important but they would not have been any greater than the differences between the Orthodox and the Catholic. They would be primarily differences of discipline: relative authority of the bishops, leavened/unleavened bread, celibacy, fasting, etc. There would have been differences in terminology. Possibly, Jerusalem and not Rome would have been the papal see, and Hebrew and not Latin/Greek combo would have been the universal language of the Church.
The Church would have remained hierarcical, sacramental and liturgical, because historical Judaism is hierarcical, sacramental and liturgical. The Scripture would have been seen embedded in Tradition and unseparable from it. Synergism between the man of free will and the Divine Grace would have been the theological norm.
That was not to be and so what we now know as Judaism is a religion that rejected not only Christ, but also the faith of the Old Testament fathers. When we look to the modern synagogue we see a mixture of the pre-Christian covenantal remnant, the loss of levitical priesthood, linked in the mysterious ways to the rejection of Christ the Eternal Priest, and layers of post-Christian theological development that is of no relevance to the question on hand.
We also have many Jews converting to Christianity and some of them are opting for one or another Protestant community, with which they blend their ethnic flavor. God bless them for that. Post-Christian Judaism has a natural affinity with Protestantism in that the former suffered the loss of priesthood through the rejection of Christ the Priest, and the latter -- through anticlericalism of Luther and his followers. To see modern Jewish converts to Protestantism as validating it is to see one error attempting to validate another.
Now let us look at the relationship that the New Testament established with the Old. In that, we clearly see a radical new light revealing its meaning. No longer do we read the Old Testament independently of the New. The precepts of the Old Testament are obeyed not because God gave them to Moses but because, and inasmuch as, Christ validated them, or reason validates them as natural law. The continuity of the Divine Revelation does not mean uniformity. Christians do not circumsize babies but they baptize them; eat pork and lobster; venerate icons and pray to Mary and saints; offer the eternal sacrifice of Christ at Golgotha and not cattle and fowl. Had the Jews accepted Christ when St. Peter preached to them, they would be baptizing babies, celebrating the Eucharist and venerating the statues and icons with us. One day, we all will.
Why establish churches and dismantle synanogues if it is one and the same faith? Because it is not the same faith!
"I believe it is close to what Agrarian was saying."
If there is any difference between what you wrote and what I said (or at least what I meant to say), I'm missing it.