The Gemara on 3a indeed asks that, as you have seen by now. Originally, on 2b, the Gemara chose Pesach and Sukkos instead, because it was looking for a "unique" Regel, as the Gemara says; there's nothing unique about Shavuos that should make it require Perishah more than the others, logically.

Later, on 3a, the Gemara accepts the possibility that Shavuos requires Perishah since it learns from the Pasuk that Perishah is required only for a single day's work, and not for a seven day Regel. This description only fits Shavuos.

By the way, are you the Yoel from Ner Yakov?

Best wishes, and apologies for the late reply Mordecai