Why didn't Christian Rome or Early Medieval Europe ever develop into an outright theocracy headed by a clerical figure in "direct communication with God?" Why the consistent separation between royalty and clergy, despite their mutual depending on each other?

Some Emperors integrated (or tried) religion more into their rule than others. Not sure how much effect this actually had but the Bible has sort of built in idea that Church and State are separate in Mark 12:17 "Give back to Caesar what is Caesar's and to God what is God's." Sometimes it is dangerous, especially when things go bad, to be the one handing down both the religious and political rulings.