The way I've always seen it, leaders are to be chosen first, and then appointed as leaders. I guess this could be compared to an inauguration. The biblical leadership model in my mind, was in some ways an indirect democracy insofar as the people choosing their leaders.
The confusion I had thought I saw was that the "choosing" and "appointment" were mixed up, and therefore church leaders choose and appoint leaders themselves.
The only New Testament scripture I can go by is Acts 6 and Acts 14. Acts 6 mentions both the selection and appointment, however Acts 14 just discusses the appointment. Many leaders I know just go by Acts 14.
Here are the scriptures I'm referring to:
2 And the twelve summoned the full number of the disciples and said, "It is not right that we should give up preaching the word of God to serve tables. 3 Therefore, brothers, pick out from among you seven men of good repute, full of the Spirit and of wisdom, whom we will appoint to this duty. 4 But we will devote ourselves to prayer and to the ministry of the word." 5 And what they said pleased the whole gathering, and they chose Stephen, a man full of faith and of the Holy Spirit, and Philip, and Prochorus, and Nicanor, and Timon, and Parmenas, and Nicolaus, a proselyte of Antioch. 6 These they set before the apostles, and they prayed and laid their hands on them.
And Acts 14:23
And when they had appointed elders for them in every church, with prayer and fasting they committed them to the Lord in whom they had believed.
There's a good article about it here which discusses the verbs used.
My question is, what does the Bible tell us about the way leaders were chosen in the early church?