“Everyone knows the קשה of the Bais Yosef, but I want to know what your answer is to the lesser known question. If טומאה הותרה בציבור then why did they need a כשר bottle of oil to begin with?”
Because that was the whole fight of the yevanim. They didn’t care if we do mitzvos or learn Torah – they cared about the fact that we believed in its kedusha. They fought against mitzvos that weren’t rational. So if we would’ve used tamei oil they would have won so to speak as they specifically defiled the oil as part of their war against kedusha and the irrationality of it. So we insisted on pure oil. And we got the nes of chanuka.