From an Evangelical Covenant perspective, as I understand it, defining the Church as a fellowship of believers stems from our experience of the Pietist revivals in the Lutheran State Church of Sweden. Before the Pietist revivals the experience was that Lutheran Church was coextensive withe the people of the Nation State of Sweden. Every Swede simply was baptized as an infant, and everyone was examined by their parish pastor about the catechism each year, this was all it meant for many to be part of the church and a Christian. Then came the Pietist revivals and being Christian became more than being baptized, attending church on Sundays and being examined about the catechism. For many this more, this new life, rubbed uncomfortable up against the formal faith of the other members of the Lutheran Church of Sweden. Many including bishops and pastors were threatened by those who came to see Christianity as something more than its formal aspects. There was a desire to regularly be with those who understood this fuller sense of being Church and Christian.
Formal definitions, the correct teaching of the Gospel and the administration of the sacraments, even apostolic succession of Bishops (The Reformation came to Sweden like it came to England.), which the Lutheran Church of Sweden technically preserved, were tied up in their minds of the Mission Friends with a State or Volk church. Questions of physical continuity with the church through time though had already been compromised with the split of the Reformation. The break the Reformation created with the structures of continuity also introduced questions whether or not there was a continuity, the question of the nature and reality of continuity through time I would suggest meant that for many continuity as a mark of the church disappear and was replaced with a belief that the mark of the church was vital and life changing faith in individuals.
When pressures within the State church pushed these pietists out into their own various denominations, for the Mission Friends who formed the Covenant there was no question that the church had to have a form and that some type of continuity had to remain. However, we concluded there was no particular form to the church and exactly where and how continuity with what came before was preserved was vague. My suspicion is that this is because on the one hand the Reformation had already pronounced some much of the forms of church and of continuity to be created by humans and thus dispensable, and on the other hand these forms were unfortunately connected with a State church.
This means that to some degree the question of continuity is not seen by the Covenant church as a central concern. Though, even as we began it wasn’t completely dismissed as we did seek to connect the ordination of our first pastors with the “apostolic succession” of the Lutheran Church of Sweden and did seek the laying on of hands from a Lutheran pastor. However, “fellowship of believers” as the central aspect of our ecclessiology is a contemporaneous a-historical understanding of the church. The Church is then made up of those who believe. Now, this can have an historical or continuous character, but the emphasis of this approach to ecclesiology isn’t upon the continuity in time, or through time, but upon the effect of enlivening faith in the individual believer is what is the primary trait of church.
My point is not to critique this particular emphasis, since I believe that the Church is not a dead work, but should be manifest in the lives of its individual members having a personal sense of the presence of God, through Christ enlivened by the Spirit. My point is that I think ecclesiology like one that says the Church is the fellowship of Believers jettisoned a strong sense of continuity because it identified aspects of Church that were in part means (symbolic and otherwise) of maintaining and expressing continuity with the State church. However, these means were not invented by the State, but were in existence before there were State Churches, they even predate Constantine and the “Constantine agreement” or Constantinianism. Further more the state church and the reality of Christendom creates the environment in which one can emphasize the church is just a fellowship of believers because the surrounding culture has institutionalized Christianity and already planted the church so that the pietist who left didn’t have to maintain the continuity with the past because the surrounding culture and State or Volk Church they critiqued preserved (for a time) that continuity. However, once Christendom and/or the stat Church disintegrates or disappears altogether the question of continuity raises its head, and shows a weakness in our ecclesiology