They certainly didn't consider themselves as ceasing to be Jewish. Sure, they were in conflict with both the religious authorities at the time, but so were the Essenes, the Zealots, and the Sadducees; so were the Hasmoneans before they became the religious authorities, and we don't suggest that any of those groups stopped being Jewish. At some point Christianity as a whole, by accepting gentiles who hadn't converted, became a new religion rather than a Jewish sect, but that isn't the same thing as the Jewish Christians apostatizing. They were in a liminal space.
The concept of Christ that we have today is, I agree, not exactly the same as any of the various concepts of Jewish Messiah that were current in say 3rdC BCE-1stC CE, but the concept(s) set out in the Pauline and Johannine corpuses and the letter to the Hebrews are responding the events surrounding the death of Jesus of Nazareth from within a fundamentally Jewish perspective. The G-d denoted by the tetragram is a G-d who acts in history and makes things new, and when that happens our understanding of the scriptures changes and they get reinterpreted. Jeremiah re-interprets Amos and proto-Isaiah and his own earlier writings when King Josiah is killed. Ezekiel reinterprets Jeremiah as he experiences life in exile. Ezra and Nehemiah and the Chronicler reinterpret pretty much everything that comes before in the light of Cyrus's decree. We can of course disagree about both the veracity of the historical events that are being interpreted, and the interpretations being made, but that doesn't mean that the other perspective is illegitimate, merely mistaken.
I'm certainly not saying that rabbinic Judaism and Christianity are equally closely descended from Second Temple Judaism. As you might expect, my knowledge of the development of Jewish thought is a lot more extensive up to about 30CE than later, but my impression is that the shift that happened within Judaism after 70CE whilst significant was not as radical a change as the development of Christianity, but it's not obvious to me that a radical change is inherently delegitimising. I suspect that the change from pre-exilic Israelite religion to post-exilic Judaism was on a similar order of magnitude to the introduction of Christianity, but as there are no extant followers of that religion, the question of appropriation seems less relevant.
no subject
The concept of Christ that we have today is, I agree, not exactly the same as any of the various concepts of Jewish Messiah that were current in say 3rdC BCE-1stC CE, but the concept(s) set out in the Pauline and Johannine corpuses and the letter to the Hebrews are responding the events surrounding the death of Jesus of Nazareth from within a fundamentally Jewish perspective. The G-d denoted by the tetragram is a G-d who acts in history and makes things new, and when that happens our understanding of the scriptures changes and they get reinterpreted. Jeremiah re-interprets Amos and proto-Isaiah and his own earlier writings when King Josiah is killed. Ezekiel reinterprets Jeremiah as he experiences life in exile. Ezra and Nehemiah and the Chronicler reinterpret pretty much everything that comes before in the light of Cyrus's decree. We can of course disagree about both the veracity of the historical events that are being interpreted, and the interpretations being made, but that doesn't mean that the other perspective is illegitimate, merely mistaken.
I'm certainly not saying that rabbinic Judaism and Christianity are equally closely descended from Second Temple Judaism. As you might expect, my knowledge of the development of Jewish thought is a lot more extensive up to about 30CE than later, but my impression is that the shift that happened within Judaism after 70CE whilst significant was not as radical a change as the development of Christianity, but it's not obvious to me that a radical change is inherently delegitimising. I suspect that the change from pre-exilic Israelite religion to post-exilic Judaism was on a similar order of magnitude to the introduction of Christianity, but as there are no extant followers of that religion, the question of appropriation seems less relevant.