Question

With God's help,
Hello to the honored Rabbi,
I wanted to ask the Rabbi a question regarding the Oral Torah—how disputes in the tradition and among the Sages came into being.
I always understood that disputes arose when there was no explicit tradition about how to act, or when a new doubt came up that had not previously been discussed. That seems to emerge from an explicit Mishnah in tractate Sanhedrin (11:2). There we see that when we have testimony about how the Jewish law is practiced, we follow it: "…they come to the one at the entrance of the Temple Mount, and he says: Thus I expounded, and thus my colleagues expounded to me; thus I taught, and thus my colleagues taught me. If they had heard, they told them; and if not, these and those came to the Great Court in the Chamber of Hewn Stone."
It follows from this that disputes can arise only about something for which there is no tradition and no explicit testimony as to how the Jewish law is practiced.
So too we see that the Tannaim adopted testimony as practical Jewish law, as brought in tractate Eduyot (1:3), where there was a dispute between Hillel and Shammai about the measure of drawn water that invalidates a mikveh, until two weavers from the Dung Gate in Jerusalem came and testified in the name of Shemaya and Avtalyon: three logs of drawn water invalidate the mikveh, and the Sages upheld their words. (And in the next mishnah it says that they did not persist in their own view and accepted the testimony.)
It follows from all this that when we have a clear tradition for the Jewish law, one does not dispute it.
 
But I have encountered a number of cases that I cannot explain—how it could be that they disputed a law that is as central as the Torah itself, and it seems very unreasonable to say that they forgot the tradition about it.
For example, in Makkot (3:10) we have an accepted law that one receives thirty-nine lashes, yet Rabbi Yehudah there disagrees and says that one receives forty lashes.
In tractate Sanhedrin (1:6), Rabbi Yehudah disputes the number of the Sanhedrin and claims that the Sanhedrin consisted of seventy and not seventy-one.
In Terumot (3:3), Rabbi Eliezer disagrees and holds that a substitute for a guilt-offering, etc., must die, against the rule transmitted in the Talmud ("we have this as a tradition") that wherever a sin-offering is put to death, a guilt-offering grazes (Babylonian Talmud Temurah 18).
In Berakhot (1:3), they disagreed whether one must recite Shema in the evening while reclining and in the morning while standing. (And it does not seem reasonable to say that this was simply common practice, since we saw the incident of Rabbi Tarfon, who reclined to recite in accordance with the view of the House of Shammai.)
 
Moreover, I saw a mishnah that could serve as an excellent example of a dispute with enormous practical implications, where in that very same mishnah testimony is given as to how it was done in the Temple.
In Pesahim (7:1) we find a dispute between Rabbi Akiva and Rabbi Yosei the Galilean about how the Passover offering is roasted. It is important to emphasize that these Tannaim lived during the time of the Temple and in their days the Passover offering was actually brought.
"How do they roast the Passover offering? A spit of pomegranate wood—they thrust it through its mouth to the place of its vent, and they place its legs and its entrails inside it; these are the words of Rabbi Yosei the Galilean. Rabbi Akiva says: That is like cooking; rather, they hang them outside it. They may not roast the Passover offering either on a metal spit… Rabbi Tzadok said: There was an incident with Rabban Gamliel, who said to Tavi his servant, 'Go roast the Passover offering for us on a grate.'" etc.
Here we encounter a significant dispute about how exactly the legs and entrails of the Passover offering must be roasted—whether inside the offering or further along the spit. What is interesting about this mishnah is that we see that these two Tannaim certainly did not disagree about the halakhic rule that the Passover offering must be roasted and not cooked. Nor do we see that they disagreed that the roasting had to be done specifically on a spit of pomegranate wood (which is dry). Later in the mishnah we see that Rabbi Tzadok testified how Rabban Gamliel roasted the Passover offering on a grate!
If so, I would be happy to know how it could be that the Sages disagreed about a law for which there surely was a tradition of how to practice it. And how did the dispute come into being?
I thought of answering that those minority opinions that we see disputing the accepted Jewish law of their time wanted to reinterpret the verses against the accepted halakhah. But that is difficult for me, because if the tradition says that this is the correct way, how does that Sage come and want to innovate against the way of the Torah?! For example, in the last case: if the accepted tradition is that the entrails of the Passover offering are roasted inside the offering, how can someone come and claim that this counts as cooking? After all, the Holy One, blessed be He, explicitly approved it… that this is not considered cooking.
I would be glad for an answer or a reference to a place that expands on how disputes in Torah law arose against the accepted tradition. For example: how did people practice until the days of Rabbi Yehudah regarding the number of the Sanhedrin… and how does the other Tanna dispute the tradition?
Sorry for the length.
Best regards,
Y. Weitzman.