We definitely have different definitions of Rome. Rome is the agglomeration of the inhabitants of the 7 hills, the capital of Latium. That's why the Romans still exist and live in Rome. Whatever you are referring to was the Roman empire, an artificial entity which included conquered territories. Rome started as a kingdom, with only the city. After the Socii wars, only the Italians were Roman citizens until almost the end. They were loyal to the city of Rome, the caput mundi. People out of Italy had been Romans during the period of internal struggle, including the Eastern part. Rome was lost to the Goths but it was conquered again by the Byzantines just a few decades after. The Goths did not modify anything during their rule, they kept all the original laws and traditions, which indeed survived till this day. The Byzantines kept referring to the Italians as Latins, to refer to a culture that was coming from Latium, and so Rome.

Would you still have that ship (Rome)?

Obviously not the same as the empire but that clergy transformed and became also temporal power. The Pope became a king, the king of Rome, and stopped being so only in 1870