the 'reverence' had nothing to do with creating the bureaucracy; the bureaucracy was created by Constantine when he placed the Christians in charge of the Roman Empire's social welfare policies. Their practice of providing welfare and social services to the members of their churches, i.e. charity, taking care of widows and orphans, etc. impressed him more than any other feature of their theology, itself derived from Jewish practices of taking care of their tribal members.
The pagan priests under the other Autarchs, particularly Lucinius, were ordered to compete with the Christians, but failed.
Constantine's success with the Christians was a big propaganda victory for him among citizens at the time. Naturally most atheist historians never mention this fact. The Christians were given a real job to do, not just sit around being religious n stuff. He didn't actually convert to Christianity until he was on his deathbed, in any case.
And, the Eastern and western Churches already had some differences early on. There were some, slight though they were, differences in the books they used as 'canon', for instance.