“Jesus was a teacher of Judaism.”
Of course He was—and more, as He claimed to be the Jewish Messiah—and rose from death after crucifixion to prove His claim.
The Covenant sealed in Jesus blood was with the Jewish people....AND extended to gentiles. The New Testament ( = “New Covenant”) book of Acts explains the integration of gentiles into the Jewish sect started by Jesus & His disciples (and that integration of gentiles was a profound struggle). The first part of Acts is almost entirely about Jews—taking place in Jerusalem, with Peter as the main figure.
Then a Pharisee named Saul—who violently oposed the Jesus-sect—was dramatically converted—and Acts (written by a physician named Luke) then follows his very adventurous missionary trips to gentiles in Asia Minor and Greece—explaining the good news of Jesus.
The end of Acts leaves Paul (Saul) under Roman house arrest, and the writer mysteriously quotes the book of Isaiah ch. 6:9-10 saying:
Go to these people and say:
You will listen and listen,
yet never understand;
and you will look and look,
yet never perceive.
27 For the hearts of these people
have grown callous,
their ears are hard of hearing,
and they have shut their eyes;
otherwise they might see with their eyes
and hear with their ears,
understand with their heart,
and be converted,
and I would heal them.[h]
28 Therefore, let it be known to you that this saving work of God has been sent to the Gentiles; they will listen!” (Acts 28:26-28)
It is clear from Acts that even at this early time (before 62 CE, when Paul was executed by Nero) — probably due to Paul’s spectacular missionary successes—that the Church was becoming more and more dominated by gentile believers—and the majority of the Jewish people had rejected the Jesus-sect...
Teachers of Judaism would never claim to be moshiah. A teacher steeped in and teaching torah would know from the get go that he did not meet the criteria for being moshiach.