Chazal (Brachos 32) record Moshe Rabeinu’s argument against Hashem threat to rebuild the Jewish nation from him alone (32:10): if a stool of three legs, meaning a nation built on the legacy of the three Avos, cannot not stand in a moment of G-d’s wrath, kal v’chomer a stool of one leg, a nation built on the legacy only of Moshe, will not stand.
The kal v’chomer invoked by Moshe seems to have an obvious flaw. The new stool will not have one leg, but four – Moshe plus the legacy of the Avos, as he is obviously their descendent!
R’ Elchanan explains (in “aggados al derech hapshat” at the end of Koveitz He’Oros) that we invoke the zechus of Avraham, Yitzchak, and Ya’akov not merely because we are physically descendent from these great people. There are other tzadikim who lived before the Avos from whom we are also descendent, e.g. Noach and Shem, but we do not invoke their zechuyos. What makes the Avos unique is not their status as our physical forefathers, but their status as our national forefathers. Just like we use the term “father of the nation” for George Washington even though he is not anyone’s physical father, so too, the Avos are the forefathers of the Jewish people as a distinct national entity. Only based on that status do we invoke their zechus.
While the lineage to the Avos would remain even if a nation were built from Moshe’s progeny alone, the Avos’ unique status as “fathers of the nation” would be lost. The new nation would be rooted in Moshe as its forefather, and would hence be a stool of only one leg.
I think this vort is interesting because we have discussed in the past in other contexts at what point geirus into a Jewish nation becomes a meaningful concept. R' Elchanan's sees the change of Avraham's name to "av hamon goyim" as reflecting a change in his status from just a family unit among the clans decendent from Shem into an indepedent nation. Yet, R' Elchanan elsewhere places tremendous emphasis on Torah as the sole binding force holding together the Jewish nation. I wonder how he squares this idea with Avraham being an independent nation even pre-mattan Torah. Of course, one could point to the Midrashim that opine that Avraham observed the Torah... just thinking out loud.