Do you know the source for the idea that Shabbat, Kashrut, and Taharat Hamishpachah are the basic requirements for being a halachic Jew?

This isn't accurate, since we don't know the rewards and importance of the Mitzvot. There are, however, several Mitzvot that are conceived as symbolic such as: Shabbat, Chametz on Pesach, fasts, pork, Yom Kippur. Perhaps Taharat Hamishpacha is because it determines the continuation of Am Yisrael. Indeed, during the generations, foreign nations forbade us these things. Rabbi Roi Margalit