Well that is what traditional Judaism holds falling blood, that when the Torah speaks of elders, it is speaking of the Rabbis. You've got to remember, the elders were in charge of teaching the people, and hence, developing the oral tradition. It is the Rabbis that pass the oral tradition and always have. That was not the function of the Priestly class. The oral tradition addresses a lot, like how to mantain the Jewish religion when there is no temple, it had to be done once before when the temple was destroyed and the Jews were exiled in Babylon. That's actually the majority of the Babylonian Talmud. The religion can be mantained without the temple by the Rabbis, and had been done before.
In this case, what traditional Judaism holds really isn't important. Traditional Christianity holds that Paul was the man who developed the Christian religion and that what we have now is passed directly from him. That is not the truth. What the traditional form of a religion says does not mean that it is historically correct.
Historically speaking, the synagogue system did not really exist until after the destruction of the Temple. There was no reason to really have a synagogue system while the Temple was in place. The reason was simply, Jewish religion revolved around the Temple.
If we look at this historically, the elders were just that. They weren't Rabbis in the modern sense. They were the older people of the community. There really is no historical evidence to suggest that they were Rabbis in the modern sense, or that the synagogue was anything like it is today.
Rabbinic Judaism evolved after the Temple was destroyed. From what we can see, it evolved out of the Pharisaic sect. However, they did not have an established synagogue system. And they were not the only Jewish sect, or even the "traditional" form. They were just one more Jewish sect during a time that Judaism was very diverse.