Was he still a priest? Yes, and in a more distinct sense than the Julian emperors were priests due to the pontifex title. So, theocracy.It's not that it wasn't purely religious, it's that I don't think it was predominantly religious.
Yes, he was also a priest. But does this matter? Al-Baghdadi and Mullah Omar were warlords, and MbS is a straight-up royal; in the Islamic world the only state with rule by actual clerics is Iran. Modern Western dystopian fiction centering around theocracy goes back and forth on whether the rulers are actual preachers or just some rando politicians - I forget which side The Handmaid's Tale is on, but in the V for Vendetta film the chancellor is a politician.
I'm objecting to adding criteria that aren't in the definition of the term, so I guess in favor of categorizing things by definition rather than by prototype or exemplar. You seem to want to exclude anything that's not a prototype or exemplar of how dystopian theocracies are represented in fiction.
Also, I would say that priests are still priests even when they are also politicians (for that matter, there are probably politics involved even when a church or priesthood doesn't claim any temporal authority outside its own organization), or even when they are venal or corrupt. So, pre-Reformation Catholic priests with incomes from indulgences and not-so-secret mistresses and families, and continually sinning and covering up or begging for forgiveness pastors like a certain Mr. Baker (sic?) or certain fictional examples, can still be regarded as priests and societies or states in which they hold the reins of power regarded as theocracies, even if not the particular type that is most prevalent in fiction. Basically, any rule by priests where the role of priest is not entirely or almost entirely ancillary to the power of the ruler(s) qualifies - so no, I wouldn't qualify the classical Roman Empire, or England once Henry made the King the titular head of the Church of England but still let that church be run mostly by the bishops so long as they didn't tick him off, as theocracies, but the Papal States even (or especially) when the Pope commanded armies, I would.