The Apostle Paul insists that civil and church authority are not completely separate, but both are established by God:
" For there is no authority except from God, and those which exist are established by God. Therefore whoever resists authority has opposed the ordinance of God;"
Second, while Christ is indisputably head over the assembly of believers, He Himself has established authority in His Church:
Matthew 28:18-20
Then Jesus came to them and said, “All authority in heaven and on earth has been given to me.
Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit,
and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.”
Matthew 16:19
I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”
Matthew 18:18
“Truly I tell you, whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.
1 Corinthians 12:28-29
And God Himself has appointed in the church, first apostles, second prophets, third teachers, then miracles, then gifts of healings, helps, administrations, various kinds of tongues. All are not apostles, are they? All are not prophets, are they? All are not teachers, are they? All are not workers of miracles, are they?
So God Himself is shown to be the one who "appoints" and "establishes" both secular and church leaders, since He is the one with the ultimate authority, and all authority comes from Him.
In no case is the authority to be exercised as "apart from God" or "against God." But if a person in authority, whether an emperor, a governor, a magistrate, a priest, pastor, teacher or father or mother, were to lead in a defiant against Christ, that person would be outside of God's will and thus one wold not be bound in obedience to follow them in something sinful.
But to say "There is never any reference to submitting to any individual over another in the assembly of God." is simply wrong. You can see the operation of authority in the actions of the Council of Jerusalem, for sure; and Paul is constantly reiterating his legitimate authority. In the New Covenant, even though Christians are a kingdom of priests (1 Pet. 2:9), Jesus calls certain men to a special ministry (Rom. 15:15–16). Christ calls men to serve the Church (2 Tim. 1:6–7) as pastors, teachers, and spiritual fathers who heal, feed, and strengthen God’s people.
That's the kind of explicitly Biblical authority we're speaking of. Not to dominate, but to be shepherds like Christ, Who especially appointed Peter, personally, to the role of chief shepherd when He told him three times to "feed my lambs," "tend to my lambs," and "feed my sheep"; and also when He called him, personally, to "strengthen the brethren."