It is not the sense in which the word was used by the Early Church.
Chapter 8. Let nothing be done without the bishop See that you all follow the bishop, even as Jesus Christ does the Father, and the presbytery as you would the apostles; and reverence the deacons, as being the institution of God. Let no man do anything connected with the Church without the bishop. Let that be deemed a proper Eucharist, which is [administered] either by the bishop, or by one to whom he has entrusted it. Wherever the bishop shall appear, there let the multitude [of the people] also be; even as, wherever Jesus Christ is, there is the Catholic Church. It is not lawful without the bishop either to baptize or to celebrate a love-feast; but whatsoever he shall approve of, that is also pleasing to God, so that everything that is done may be secure and valid.
This is the same letter that condemns those who don't believe in the Eucharist as the Church teaches it, and condemns all schism. When he wrote "Catholic" he meant "Catholic".
Call it what you will. It won’t change the fact that catholic means universal and all believers are part of that universal church. We are not Catholic, we are catholic. All of the early church was not Catholic as you put it.