No, it doesn't. You seem to be thinking that they would derive such legitimacy purely from geography, but that is blatantly false. The HRE had Italy, it lost Italy. Its desire to reclaim Italy had nothing to do with lacking Legitimacy once it lost the Italian territories, and the fact that Northern Italy was tearing itself apart with pro- and anti- HRE factions would argue strongly for its legitimacy, with its lack of martial ability being its greatest failure. The Eastern Roman Empire's desire to reclaim Italy was never about Italy not being part of the Roman Empire, Italy was considered part of the Roman Empire - and thus the Byzantine Empire - before Justinian invaded. Justinian's invasion was rather to suppress a parallel political authority independent of Constantinople. It was always considered Rome and Roman, and the Italian populous knew this well from their own very accounts. Constantine, Otto, and Mehmet are very different men and derived legitimacy in very different ways. The more Islamic their empire became, demographically speaking, the more they deemphasized their initial connection to the Romans. Their play for Italy had little to do with the Roman Empire, and far more to do with the Pope and the Sultan's own position as Caliph.