"Well into the writings of the Early Church Fathers they were still referring to themselves as "The Church of God". You can wish and hope as much as you want....but the term "Catholic" is a "Johnny Come Lately"."
You've got your history wrong there, Diego. No later than 107 AD, +Ignatius of Antioch decribed the structure and visible community of the "Catholic" Church in his Letter to the Smyrneans. He was the second successor to +Peter at Antioch and a disciple of the Holy Apostle John. Its likely the term was around substantially before 107.
Polycarp, disciple of the Apostle John, referring to the Church of God well into the first century. See the first line.
Another example The first epistle of Clement to the Corinthians...2nd line.
There are many other examples of the Early Church Fathers writing "To The Church of God, from the Church of God," etc.etc.
I would also argue that the term church of God is a descriptive term and not the same as Church of God a proper name.