There is a difference between “catholic” and “Catholic.” The latter is used to describe a denomination. The former is used to mean “universal.” It is similar to the denomination, “Church of Christ.” Every Christian is a member of the church of Christ. It has always seemed rather arrogant to name a denomination, THE Church of Christ. The issue is that the Roman Catholic Church does not make a distinction between small “c” and capital “C” catholic. Only those that are in communion with the Bishop of Rome are considered catholic.
No Biblical designation “Church (singular) of Christ.”
There are “churches (plural) of Christ.”
No one worldwide, man-led organization necessary.
There is (1) the Body of Christ (the spititual union of all genuine believers in this age in one Head, Jesus Christ—the family in heaven and earth, not just earth); (2) individual local assemblies which are admonished to mirror that spiritual union.