it is fair to say that the turks wanted to  be romans, but the romans did not want to be turks. so, the turks tried to model themselves after rome as a successor state, but what was left of the population got up and left - some of it returning to rome (where it kickstarted the renaissance) and some of it fleeing north to russia. i hold to the historical idea of moscow as the third rome, and see the russian empire as the continuation of byzantium. as such, the fall of constantinople was not the end of the romans, exactly - so much as the loss of the capital to the persians. empires shift over geographic boundaries and ethnic groups, but very few are ever truly destroyed.