The ultimate answer lies not with the Rabbis but with the Christians. Specifically with Paul/Saul of Tarsus and finally the Christianized Roman Empire.
To understand the issue you need a good background in Church history.
First let me clarify the etymology of the name Jesus. In Hebrew, Jesus' name would be correlated to the anglicised Joshua which is actually Yehoshua יהושע (LORD is Salvation). In the Torah, Moses had renamed Joshua's original name from Hosea ושע to Joshua יהושע. But in Greek, in which the Christian testaments were consolidated, Yehoshua was transcribed due to Greek grammatical declination as Iesous, which became latinized into Iesus and then romance languages furthered it to "Jesus" where the J was originally pronounced with the consonantal Y.
The messiahship and deity of Jesus
Many Jewish commentators believe that Jesus/Joshua never claimed to be a deity or the messiah. However, that is irrelevant to the question. It does not matter whether Jesus had made such claims - what matters was what his followers and early adherents had ultimately made him to be.
From my analysis of literature, early Christians had frequently sought refuge in synagogues from pre-Christian Roman persecution. And in doing so, attempted to proselytise their Jewish brothers. This practice became intolerable and was the beginning of the rift between Jews who believed in Jesus and those who did not.
Pauline theology
Peter had sought to keep Christianity a Jewish domination. Paul, as you can read from the epistles that he (supposedly) had written, invented a new twist in the universalization and globalization of the Christian gospel. Paul famously wrote to the Roman church that salvation is by faith and not by works. Which directly countered the theology of James and Peter. In that, Paul asserted that believers in Jesus should no longer need or seek to fulfill the covenant of circumcision. That was the most brilliant marketing strategy that Paul had to make for the new religion he invented to be completely palatable to non-Jews. Within the power struggle, Paul is believed by many analysts to have appointed himself as the apostle to replace Judas - even a replacement had already been appointed by the auspices of Peter's circle.
The globalization of Christianity had a very pronounced effect on the Roman empire. The number of disenfranchised Romans (and non-Romans alike) had become too numerous, and had found Christianity as the suitable replacement for the degrading morals and morale of the Roman Empire. Did Emperor Constantine really had a vision or he saw the wisdom and opportunity of reuniting his empire under Christianity, or both - is left to be guessed. Christianity was no longer a Jewish sect. Paul's legacy had triumphed over Peter's.
While Paul was the foundation of the Roman church and that the Roman church owes its existence to Paul, it is ironically Roman Church tradition that Peter (the theological adversary of Paul) is the founder of the Roman Church.
The Christian Church had become overwhelmingly non-Jewish.
Persecution of Jews
Due to Jewish intransigence in accepting Jesus' deity and messiahship, the now powerful church began persecuting Jews and started blaming Jews for crucifying their founder. Especially where many of the Christian testamental scripts depicted Jewish priesthood and commoners as the enemy of Jesus. There was no unification possible between Jews and Christians at all from then on.
The observance of Sabbath
Early Christians who were Jews still observed the traditional Jewish Sabbath. Therefore, they had to meet on the morning of first day of the week, in performing their "work" of furthering their gospel message. So meeting on Sunday was considered "work" that should not be performed on Shabbat.
In my personal theory, non-Jewish believers consequently mistook the Sunday meetings as the Sabbath and evolutionarily and gradually instituted Sunday as the Sabbath.
There were reports that the rate of conversion to Judaism competed with the rate of conversion to Christianity in the Roman Empire.
The Church finally put its foot on the ground and declared that Sunday is the new Sabbath of the "new covenant". This had the effect of helping people differentiate which religion they were actually converting to or had converted to and that the two instruments of faith were not inter-convertible.
The declaration, I believe, was a great relief to the rabbis.
The current status
Modern Orthodox Judaism would still recognise Jews who had converted to Christianity as Jews but not the state of Israel.
In Israel, I have encountered some Jews who declare their belief in the messiahship of Jesus. The legality of their Jewishness being recognised by the state of Israel is intact because they do not declare themselves as Christians.
A minority of them believe in the messiahship of Jesus but not in his deity, which should not be problematic because after all many Chassids in the Lubavitch circles believe that Rabbi Schneerson is the messiah. Legally speaking, what is the difference anyway?
What I do find concerning is a large number of Russian Jewish immigrants who are legally recognised as Jews, but who have become very accustomed to celebrating Christmas in the land of their birth. And who have become very persistent in abolishing the practices of Shabbat and the abstinence from a non-kosher diet in the legal and municipal regulations.
But on the whole, Christianity is not accepted as a Jewish sect at all primarily due to the belief in the deity of Jesus. Even then the theology instituted by Paul have made it impossible to be considered a Jewish sect.
