On the contrary, the Church was named “Catholic” because it already was catholic in fact. Here’s what I mean: In the New Testament, we see letters written to this church and that: The church in Corinth, the church in Ephesus, etc. In Revelation the angel is told to write messages to the church in this place and that place. Yet the New Testament also says there is one body, one faith, one baptism, etc. (Eph. 4:4-6). In other words, one Church.

The name “Catholic” was simply a way of referring to the One Church, as opposed to a church in a particular place. The Catholic Church was the one that encompassed all the local churches. The local church at Ephesus was not itself the Body of Christ, but an assembly of some members of the Body, of which there were many members and many assemblies. The Body on the other hand was the Catholic Church, with Christ as its head, of which there was only one.

But almost from the beginning there were those who taught otherwise than as the Apostles and Jesus had taught. The early Christians were told to shun such teachers and have nothing to do with them (Gal. 1:8). These early heretics and schismatics considered themselves true Christians and, perhaps, members of the Body. But it was not theirs to decide whether they were truly Christian and truly members of the Body; the Apostles made that determination, based on whether they were obedient to the traditions which they themselves had handed on (2 Thess 2:15).

So from the beginning you have rival groups preaching different gospels, yet claiming to be true followers of Jesus. Some of them were even baptized Christians. Does this make the rival groups members or parts of the true Body, the Catholic Church? Did the Apostles welcome their false teaching for the sake of diversity of opinion? Clearly the false teachers were rejected, and furthermore repeatedly warned against throughout the New Testament. This in fact is the origin of the phrase “a wolf in sheep’s clothing”.

So now you have the Catholic Church, the true Body with Christ as its head; and then you have other groups claiming the name of Christ, but which are not members of the true Body. Notwithstanding their claim to the name of Christ, they are not members of the Catholic Church. Is this exclusion due to intolerance on the part of the Body? Is it the fault of the Body that these rival claimants cannot truly be called members of the Catholic Church? Or is it their own fault, for preaching a gospel different from the traditions handed on by the Apostles?

Now tell me: Would the Apostles have invited these false teachers, and their followers, to partake of Communion with the members of the true Body? Obviously not, since they had already instructed their flocks to shun false teachers. How could you shun them and yet share communion with them?

Thus the exclusion of certain people and groups from the Catholic Church, and the denial of communion to their members, is not the result of a failure to appreciate diversity on the part of the Church. Rather it’s the result of the refusal of these people and groups to submit to the traditions which have been “handed down by word of mouth or by letter”. It’s they themselves, or their forebears whom they choose to continue following, who made the decision to preach a different gospel. The Church in shunning them and denying them communion, is only obeying the warnings and injunctions of which the New Testament is filled.

Your premise seems to be that any person and any group claiming to be Christian and members of the One True Body, with Christ as its Head, has the right to demand acceptance, inclusion and communion, and the name “Catholic”, merely by virtue of claiming to be such. To me this is not only contrary to Catholic teaching, but clearly unbiblical.