Yeah that is true a theocracy is more like the Papal States or Iran with the Ayatollah. Byzantium if you do your homework had a state church but was not at all a theocracy. It was more like an anarcho-monarchist state with a non-feudal administrative system like China. The throne was literally up for grabs by anyone who wanted it, you just had to oust the emperor which is a throw back to Ancient Greece and Rome. The Orthodox Church, the codex justinianis, and a formal bureaucracy kept the world turning while Emperors were doing whatever. The west had hereditary rule and feudalism. Americas government doesn’t even have God in the equation - hence the question. You could have a federal system with a “church of America” or a “Christian supreme council” like how they have different councils and caucuses in congress.
No, you forgot an important part of the equation. No one could be a citizen of the State unless they were a member in good standing of the Church. Hence if you were excommunicated for a certain length of time, you lost citizenship.
Naturally, Talmuds never had citizenship, which mean they couldn't serve in the government, schools, banking, or official media.
Do you have proof or source towards Byzantium being a theocracy?
It's basic knowledge, easily googlable or wikipedia'd. The gateway to citizenship was through the Church, which gave the Church a massive amount of institutional power. The Emperor system was insanity, but in spite of this the Church literally ran the state by keeping out those who hated Christ.