"While there appears to be some order, no one voted for leadership."
I think you'll find that some bishop always presided at these early synods and its likely they were elected. Eventually, the bishops of the major cities came to regularly preside at local or provincial synods and later the Patriarchs of the major Sees of the Empire. But, at least from an Orthodox point of view, this primacy, while invested with such authority as necessary to make the primacy real and effective, was still a primacy among equals. We still operate this way.
When the apostles sought to replaced Judas they cast lots for his replacement; they did not "vote" for his replacement. The fundamental difference is they left it in the Lord's hands knowing they were incapable of understanding who would be the best choice since they cannot see into the heart. It is unclear how people were selected 100 years later but in the very first beginnings of the church, they were not elected. Now if you want to follow tradition you need to get some dice. ;O)