Ah, no. Byzantium is the Latin version of the Greek Byzantion, a city named after King Byzas. Constantine remaned the city Nova Roma, but it was popularly called Constantinople. You are however correct about the etymology of Istanbul.
OK, still before it was destroyed by the Seljuks, it was Byzantine Empire with Constantinople the capital of it. Most cities and countries have a history of names, but justice called for the restoration of names and conditions of that which was subdued and destroyed by force.
If instead it should be restored as Byzantium, that would be fine by me, but it probably will take up the name it last had.