The church of Christ has always had leaders and a teaching ministry (cf. Eph. 4.11, Titus 1:5, Heb. 13:17, etc.). It’s not wrong to think of the church as an institution; the church does have a structure. But the church also is the body of Christ, a living community that is called to love and serve. So we can also think of the church as an organism. Herman Bavinck explained this in a balanced way:
Government is indispensable for the church as a gathering of believers. Just as a temple calls for an architect, a field a sower, a vineyard a keeper, a net a fisherman, a flock a shepherd, a body a head, a family a father, a kingdom a king, so also the church is unthinkable without an authority that sustains, guides, cares for, and protects it.
Bavinck then explained the role of Christ in governing his church. He goes on:
…The church is not conceivable without a government. Granted, Christ could have exercised his office without any service from humans. If it had so pleased him, he could have dispensed his spiritual and heavenly blessings without the help of institutions and persons. But this was not his pleasure; it was his pleasure, without in any way transferring his sovereignty to people, to nevertheless use their services in the exercise of his sovereignty and to preach the gospel through them to all creatures. And also in that sense the church was never without a government. It was always organized and institutionally arranged in some fashion.
As the gathering of believers, the church is itself used by Christ as an instrument to bring others to his fold. By it Christ administers his mediatorial office in the midst of the world. Thus, from the very beginning, the church appears on the scene in a dual form. It is a gathering of the people of God in a passive as well as an active sense; it is simultaneously a gathered community and the mother of believers or, in other words, an organism and an institution.