I don’t think B”H and B”S are just lenient and strict. This is like saying that Hillel was a rabbi for a less observant congregation. I heard a deeper pshat that had something to do with hillel believing in being a realist, and following halacha at your level but shammai held that one should constantly aspire to higher levels. This is why Shammai would set aside food for shabbos during the week, aspiring for the kedusha of shabbos while Hillel would porposely rely on Hashem to provide the necessities for Shabbos in time. This follows the idea that we deal with what we have and trust Hashem on the rest. This is why when Mashiach comes the halacha will be like B”S since we will then be holding at that level he aspired to reach.
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