If the muezzin is not heard until the end of the adhan, does the listener begin to repeat the adhan from the beginning quickly and then continue with the muezzin, or is that incorrect?

If you miss part of the adhan, do not recite it. Instead, respond to the muezzin from the point you hear him, because responding to the muezzin is contingent upon hearing him, as indicated by the Prophet's (peace be upon him) saying: "When you hear the muezzin, say as he says." This is affirmed by An-Nawawi and others.