Why did the German emperors strive to be crowned in Rome?

The first coronation of the emperor in Rome (during the Middle Ages) took place in 800. Then Charlemagne was crowned under the title “Emperor of the West”, which hinted at succession from the Western Roman Empire (existed from 395 to 476, and as a single empire for 400 years longer). It was a kind of “counterbalance” in Byzantium Europe, where the emperor inherited the title from the time of late antiquity. The Pope was also in Rome (in fact, the patriarch of Western Europe), it was he who crowned the emperors of the Holy Roman Empire since 962.