The Baptist churches of the 17th and 18th centuries developed some patterns and terminology for church planting that has a lot of wisdom and worked very well. This pattern continued to be the basic pattern for Baptists through the 19th century but slowly diminished during the 20th century and is not heard of much today. That includes the idea of starting one church out of another by sending out an "arm." This arm was usually a small group from the old church that wanted to start a church in another place. I think this is a pattern that we would be wise to reestablish today.
However, a good pattern can become a terror when it is adopted as an unchangeable rule and used for control over others. It is certainly a biblical pattern to start new churches out of existing churches. We see this in the sending of Paul and Barnabas as missionaries (Acts 13:1-5). They went as sent out from Antioch and established churches in many places. However, to say that only churches that are started in this way are biblical is to turn the principle on its head. It requires a kind of "apostolic line" for churches and puts the authority for a churches existence in its history and not in the word of God. It also calls into question the "New Testament" status of every church. What if the church out of which you were founded was not an "arm" of another church? Despite a couple of churches that claim to know their heritage, no one can know for sure how their line has continued for the last 2,000 years.
Making this good principle into an absolute rule denies the possibility that God can ever start a church in another manner. What if the church out of which you came goes into apostasy and a minority group in the church rejects this apostasy? It is unlikely that the existing church will gladly send the minority out as an arm of their apostate church. Is there not any higher authority to which they can appeal when the church has gone bad? Certainly there is. That authority is found in the headship of Christ and the authority of the Bible. There are times when a group decides that they must obey God and not man (Acts 5:29). So, although the founding of a new church by sending out people from an existing one is a good biblical pattern, it is not a rule of terror that makes any other founding of a church non-authoritative. The final authority for the church is found in scripture and the Chief Shepherd of the church is Jesus Christ. We should not lightly bypass the biblical pattern (as I see many doing today), but we must not make an oppressive rule out of it either.