I have a Question: for Yevamos 15a, near end of Amud, where the Gemara says:
Question (Mar Zutra): It occurred that Shamai's daughter-in-law gave birth on Sukos. He enlarged the Sukah to make room for the baby. This shows that Beis Shamai followed their own teachings!
My questions are as follows:
(a) The only reason to think that Beis Shamai did not follow their own teachings was because Beis Hillel were the majority. This does not apply to Shamai himself! Once, Hillel himself was Kafuf to Shamai (Shabbos 17a). Hillel did not admit to Talmidei Shamai that he was bringing Olas Re'iyah on Yom Tov (Beitzah 20a). Surely, the Gemara's question was about later Doros!
(b) Also, since they were arguing for years, and each side said 'the Halachah follows us', surely Beis Shamai were stringent for their own opinion. It seems that the question was only whether or not they were stringent to be concerned for Beis Hillel's opinion!
(a) Perhaps the case of Sukah was not a Machlokes of Hillel alone with Shamai, but of other Tana'im as well. The Mishnah in Sukah does not say who the Cholek is, but quotes them Stam, from which we may infer that they where a majority. The Ritvah refers to them as "Rabanan".
Alternatively, the Gemara in Shabbos that says that Hillel was Kafuf to Shamai, is not a general statement, but refers only to that specific Halachah in which Hillel was Modeh to Shamai. This is apparent from the Gemara Shabbos 15a, which infers from the fact that the Beraisa says that Hillel was Kafuf to Shamai, that Hillel was Modeh to Shamai. In the case of Sukah, Hillel was not Modeh to Shamai. Consequently, even Shamai would be obligated by the rule of Achrei Rabim l'Hatos. The Gemara in Shabbos 17a says specifically that Hillel was Kafuf to Shamai "on that day". Accordingly, we must say that the story of Hillel in Beitzah, where he avoided admiting that he was bringing an Olah was not because of him being Kafuf to Shamai, but merely because he wished to avoid a controvesy. This Pshat is supported by the fact that Tosfos in Chagigah 7b points out that Hillel in fact never argued with Shamai himself about bringing Olos on Yom Tov; the Machlokes was only with his Talmidim. One assumes that Hillel was not Kafuf to Talmidei Shamai.
(b) The Ritvah (along with other Rishonim) concurs with your suggestion that even according to the opinion that Lo Asu, they were still Machmir for their own Shitos. He asks your question, and replies that the Gemara means to ask that according to the opinion that Lo Asu, even though l'Chumra Asu, nevertheless, where it was possible to be Machmir in private, they would not have done so publicly. (This indicates that the Ritvah understood that the reason for Lo Asu is because of Machlokes.)
Tosfos and other Rishonim however hold Lo Asu, even le'Hachmir (in other words they were even Meikal not like their own Shitos). This may be possible because once the Torah forbids Agudos, we see that it is Matir the minority to be Meikal like the majority. (If so, Tosfos understood that the reason for Lo Asu is because of Lo Sisgodedu.)
Dov Freedman
I would be satisfied with this answer if all agreed that me'Ikar ha'Din the Halachah follows Beis Hillel, and we ask if Beis Shamai were stringent for their own opinion because they were sharper. However, each side said 'the Halachah follows us' (Eruvin 13b). This does not connote 'our Drashos or Kal va'Chomerim...are more reasonable'. Rather, it seems like R. Yochanan explained (Yevamos 14a) that Beis Shamai thought that the Halachah follows themselves because they are sharper, and (most of) Beis Hillel are not counted in the Minyan, so Beis Shamai are the majority!
Pesach
As far as I understand your question, you are asking according to the opinion that Lo Asu according to Tosfos. The Gemara offers two explanations for this opinion. Either that Beis Shamai followed the opinion of Beis Hillel because Beis Hillel were the majority, and Beis Shamai did not hold that being sharper mitigates being this, or because of the Bas Kol. Either way the Halachah was fixed like Beis Hillel.
As for the Gemara in Eruvin, we must say either that it was at an earlier juncture, or that they meant that the Halachah ought to be like us.
Dov Freedman