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Another way to look at it - Why did Christ institute “the church” on earth? Obviously first is that He might be glorified in/through her.

But second, the Church was to “go, make disciples” - baptizing and teaching... How can any invisible nebulous entity do that?

The way that Catholics understand the church always confounds me. It's like they have never considered the trinity. There is one God in three persons. There is one one church in many churches. The church in Rome is the church. The church in Corinth is the church. But just as the father is not the son, so the church in Rome is not the church in Corinth.

From the Athanasian creed "So likewise the Father is Lord; the Son Lord; and the Holy Ghost Lord. And yet not three Lords; but one Lord. For like as we are compelled by the Christian verity; to acknowledge every Person by himself to be God and Lord; So are we forbidden by the catholic religion; to say, There are three Gods, or three Lords"

The same understanding applies to the church. In Revelation, the seven churches in Asia are listed. Each has a lampstand, with Jesus standing in the middle. The church in Ephesus "tested those who call themselves apostles and are not, and found them to be false". The church in Ephesus did this on it's own. Their own lampstand; their own judgement. When the scriptures say to take a problem to the church, they are not talking about some singular earthly institution that covers all believers, they are talking about the local church, which is the church.

A very thoughtful and useful insight. Thank you.

Here is Paul's description of what drives Christian unity:

Eph 4:4-7 There is one body, and one Spirit, even as ye are called in one hope of your calling; (5) One Lord, one faith, one baptism, (6) One God and Father of all, who is above all, and through all, and in you all. (7) But unto every one of us is given grace according to the measure of the gift of Christ.

It is not "One hierarchical organization, one Pope, one Magisterium" to rule them all. What makes all believers one Bride of Christ is our one glorious Bridegroom, Jesus Christ, being magnified among us by one Holy Spirit, all to the glory of the One God. And as there is love between all three Persons of the Trinity, so there is love between all of us who believe, for each other, and for our Christ, our Holy Spirit, and our Heavenly Father. This is what Christ said would mark us as His very own, our love for each other, and how else could it be, as it is the inescapable reflection of our union with the Triune God.