Hillel, Bnei Beseira and Passover

Meiri, Introduction to Avos (Seder Ha-Kabballah, pp. 54-57):

The story is told in the Talmud (Pesahim 66a) that the elders of Beseira were the nesi'im (religious heads) in the land of Israel and were known as leaders in Torah teaching. It is said about them that once they forgot whether the Passover [sacrifice] take precedence over the Sabbath or not, whether this doubt was at it simply seems, or it has some secret hidden within it, or they forgot how to extract this matter through Talmudic exegesis. They were forced into this by someone who argued against the practice, and Bnei Beseira could not conclude this matter. They were embarrassed and ask if there is anyone who knows whether the Passover [sacrifice] taked precedence over the Sabbath or not, i.e. does anyone know how to respond to this arguer with conclusive arguments from the Torah, whether from analysis, comparison or from testimony of a tradition. They were told that there a certain man who had come from Babylonia, Hillel the Babylonian by name, who had studied under the two greatest sages Shemaiah and Avtalyon, and perhaps there is hope from him. They sent for him and he came. They asked him, "Does the Passover [sacrifice] take supersede over the Sabbath?" He said to them, "Does only one Passover [sacrifice] take supersede the Sabbath? Are there not more than 200 Passover [sacrifices] that supersede the Sabbath each year?" i.e. the [Sabbath] sheep sacrifices that are similar to the Passover sacrifices, of which four are brought every Sabbath... They said, "Did we not say that we would have from you hope? But from where do you know this?" He proceeded to expound from comparison, kal va-homer (a fortiori) and gezeirah shavah...

However, it is said there that they disagree with his proofs ... Even though he argued with them all day, they did not accept it from him until he said, "Let it be brough upon me, this is what I heard from Shemaiah and Avtalyon." Immediately, they accepted it from him, sat him first and appointed him the nasi.

In other words, they knew that the Passover sacrifice supersedes the Sabbath but could not prove it. This frustrated them immensely, and Hillel attempted to prove it to them. However, they did not accept any of his logical proofs and only accepted his testimony that he had a tradition from Shemaiah and Avtalyon that this was the halakhah.

This is not, of course, the only way to explain this exchange.