Well, the early ones did. Charlemagne was crowned by the highest law he recognized, the Pope, who was from Rome. Plus the successors of Charlemagne were the descendants of a Romanized German. Little was known of how Byzantium was doing shortly after Charlemagne, but there was fierce competition that resulted in the Great Schism of 1054. After that, they used the excuse of the Byzantines Orthodox church.

tl;dr Some earlier emperors probably did, but after a few hundred years the feeling passed.