While this is not a perfect answer to your question, I suggest you give it a read (all credit to u/blacktiger226): https://www.reddit.com/r/AskHistorians/comments/787k2d/why_was_mehmed_ii_so_obsessed_with_rome_and/

There is great propaganda value in claiming to be the Roman Empire. When the Western Roman Empire began to decline precipitously and eventually fell, the Eastern Roman Empire (known by historians as Byzantine, although this was not an official name) held on to the title and were actually quite angry when the Pope crowned Charlemagne Holy Roman Emperor. Thus is the eyes of many Constantinople was a second Rome, and in previous times, a greater Rome than the first. When Mehmed takes Constantinople, he actually entitled himself Kayser-i Rum, which means Caesar of the Romans and he wanted to position the Ottomans as the natural successor of Rome, to give the Empire legitimacy and prestige.