I think the explanation of Rashi’s second peshat is simple-“Rubo K’kulo=since the majority is bain adam l’chaveioro so Hillel considered it a good answer to defining “Kol” Hatorah Kulo.
With Regard to the words of Hillel, I like Daas Yochid’s explanation and would expound on it. The essence of Kabalas Hatorah was that Klal Yisroel then became Kaviyochol “One” with Hashem through the Torah; as it says in Zohar “Yisroel V’Orayasa V’Kudsha Brich Hu Chad Hu”= Hashem, the Torah and Klal Yisroel are “One” spiritual entity with Hashem kaviyachol. This is what Rashi in Parshas Yisro on the posuk “V’Yichan Shum Yisroel Neged Hahar” says Rashi-“K’ish Echad B’laiv Echad”, meaning that the essence of Kabalas Hatorah was a unification of all Klal Yisroel with Hashem through the Torah to become kaviyachol “One” with Hashem.
So Hillel is saying that the essence of the whole Torah is this unification concept, and the Taryag mitzvos are the mediums with which to ensure and maintain this unification with Hashem. This is why Hillel used the words “that which you dislike being done to you, don’t do to your friend”, meaning this concept of “unification”; if you transgress a commandment of the Torah [be it bain adam l’makom or bain adam l’chavero], you are actually severing your own connection with Hashem and your fellow Jew.