“How does a purely invisible church adjudicate between two earthly people who have a dispute?”
I didn’t say I thought the church was purely invisible, just that it is a primarily spiritual organization, and not a physical one. Obviously, the spiritual church has a physical presence on earth, since many members of the church are physically living on the planet at any particular time. The problem is mistaking that physical presence of a spiritual organization for a physical organization, and then giving the physical organization a greater prominence than the spiritual.
In the NT, the spiritual things are always placed on a higher level than the physical, for example, when speaking of the nature of sins, or what we should value as treasures. So, it is clear that the spiritual church, of which Christ is the head, is superior to any physical organization that Peter may have been the head of, or any of its descendants.
A false dichotomy. Christ said to Peter, "And on this rock I will build MY Church." Christ's church IS Peter's church. That's what Christ said.
Just like a human being is both spiritual and physical, Jesus founded upon the rock of Peter a physical and spiritual Church, and told people to go to it when they have a grievance. You can't "go to" an invisible church with your brother to get an authoritative ruling on your dispute. A Church has to be visible to bind and loose in Christ's Name.
While Jesus is only recorded using the word "church" 2 times in the Gospels, "Church" appears 70 times in the Epistles. Here are the things that the Church is told to do:
These are legitimate churches which make up the Church. They have successors. This is a continuous process which has never ceased, in all of its many conflicts and travails never suffered a total irreparable rupture, in any century since the first, and to this very day.