How did those Jews in Jerusalem break the covenant? That story started long before the time of Jesus. After the successful second-century BC revolt of the Maccabees and the reestablishment of an independent Jewish state, the Hasmonean kings (from the Maccabee family) eventually assumed not only the kingship but also control of the high priesthood. The king and high priest became one.
The Zadokites among the Essenes considered the non-Zadokite priests usurpers and declared their Temple sacrifices illegal. Although they could not function as priests in the Temple anymore, they followed the holiness standards for priests in ways deemed more authentic to the Torah. They separated themselves from the mainstream to form exclusive communities of the righteous. They studied the Prophets carefully and looked for God’s impending judgment of Jerusalem’s unfaithful leaders. They believed their community was the beginning of a renewed people of God in earth.
Jesus held many of the same beliefs. It has led some to theorize that Jesus himself should be called a Separatist, or even equated with the Teacher of Righteousness who led the Qumran community. But we have to look more closely at the Separatist communities to see where Jesus agreed and where he called for an even more radical way forward.