“You have a confused understanding of the Church.”
As I have said many times before, the church is made up of people who have believed in Christ and, as a result, God has supernaturally, through the work of the Holy Spirit, transformed each one to be member of the body of Christ, the church. Simply put, the church is believers. It is nothing more than people who have trusted in Christ, and nothing less. It is not hymnals, pews, steeples, stained glass windows, organs, catechisms, artifacts, relics, altars, buildings, or even Bibles. It is people.
Some of these people are alive on earth, imperfect and still in our mortal bodies. Some are in Heaven, perfected but waiting to be clothed with immortality.
The word “church” essentially means congregation, i.e. a gathering of people. What do congregations do? They congregate. A local church is a group of believers who gather together.
“Why would the Apostle John be under the impression that local Churches had to recognize his authority?”
Because he is an apostle. His words are still authoritative. Believers (i.e. the church) reads them daily for instruction and encouragement.
I have given you my definition of the church. Now tell me yours.
So you apparently think that authority and organization in the Church perished with the last apostle. The Church, on the other hand, believes that the Apostles appointed successors — namely, the Bishops, and that these bishops continued to have authority over the Church. You can read about this in the Epistles of St. Ignatius of Antioch, a disciple of the Apostle John. And for my on my view of the Church, see “On the Unity of the Church,” by St. Cyprian of Carthage, who was martyred in about the year 258:
http://www.ccel.org/ccel/schaff/anf05.iv.v.i.html