Someone has ruled over pretty much everywhere, but that's not the point. The Russian claim is not the same thing. I'm not talking about a hypothetcial legacy. I mean literally, the Catholic Church is the unbroken continuation of part of the Roman state all the way back to the republic. That's a fact. It is the same organization with a continuing sequence of office holders. Even the still used term "pontiff" predates Christianity. Only the gods have been swapped out. When the political components of the Roman state were wiped out the religious component survived and still exists. The Roman state never entirely went away.
Yes, 110 acres are there. Some of the terminology is there. Pontiff is derived from the same root as pontoon, meaning bridge.
However the seat of government had shipped out of Rome aeons before Leo the Great and Gregory the Great increased Church involvement in what had been state affairs, but it was more along the lines of taking over the ancient equivalent of the EBT system.
Formal sovereignty came with either Charles Martel or Charlemagne conquering the area and then giving things to the Pope. Up to that point Constantinople still had a nominal claim for sovereignty.
At any rate, even if one grants your claim, the Jewish state now dwarfs the Roman.