Careful, though; the question is* does* the* title come with authority of its own*?* I do not think it can be so*, especially historically. Surely the bishops of Alexandria had the same authority on account of their Orthodox faith before St. Pope Heraclas. In modern terms, of course we say “the Pope” (meaning for us, the bishop of Alexandria) has authority, because “Pope” is the title of the holder of the office of highest-ranked bishop of the Orthodox Church of Alexandria. But if we look beyond that, and ask ourselves “well, why then is he the highest-ranking bishop?”, we see that it is not essentially different than why any lesser bishop, or priest, or other person holds the rank that they do in the church: He holds the Orthodox faith. The difference in authority comes in what he is called to do with it, as a representative of the church on a larger scale than the average person. It does not come by virtue of the title, because indeed the office was not always thought to be one with the title (as is even more so the case in Rome, since the bishops there did not take up this title until several centuries after it was bestowed upon the Bishop of Alexandria).
Where is the authority? With the Orthodox bishops, priests, monks, etc. How do they get to have that authority vested in them? On account of their Orthodoxy which is tested and proven so as to make them worthy candidates to be bishops, priests, monks, etc. in the first place. So it has been at some times in history that the Alexandrian Pope has been deposed, same as in the case of the* Roman Pope, for his infidelity to the Orthodox faith*. The faith is paramount, and next to it the title is nothing.