As someone else pointed out, the term "Catholic" was not originally identical with the Roman Catholic Church. When we say the Apostles or Nicene Creed, the term "Catholic" did not mean what is usually implied today, when they were developed, well over a millennium before the Reformation.
From Wikipedia -
So when the OP asks about "The True Origin of the Catholic Church", what he really means is what is the true origin of the Roman Catholic Church.
For a start the term wasn't even used until the Reformation, when (again as someone else has pointed out) it was used initially as a derogatory term, just as "Hot Prot" or "Cattle Tick" might be used today.
The (Roman) Catholic claim about itself goes back to the debate about the Papacy when it's all said and done.
There are only three contenders as the "original Church". The Protestants don't even get a look in, as they broke away from the Catholic Church 1500 years after Christ gave the imprimatur to Peter, and in doing so set up an office.
The problem for the Orthodox Church is that it derives from the Byzantine Church, which was headquartered in Constantinople. The trouble with that is that the Holy Roman Emperors assigned themselves the role of church head, which had nothing to do with Christ's declaration to Peter, who went to Rome, and took on a bishopric role. Then in 1453 Constantinople was sacked, and it became Istanbul, a Turkish Moslem city. The last emperor Constantine XI perished in the fighting (although there is some uncertainty about how he died), and the Holy Roman Empire centred in Constantionople came to an end.
I used to wonder why Constantine was so willing to move the Empire's seat to Constantinople. It appears he was born in what is now Serbia, and had prior experience in living in the general area. He also was largely free of attachment to Imperial Rome itself, and was therefore more willing than most to make the move. So the Empire's governance effectively moved from Rome to Byzantium, or Constantinople as it later became known. The secular Roman Empire centred on Rome became the Holy (Christian) Empire centred on Constantinople.
Rome became a backwater for quite some time variously sacked by Vandals, Huns, Goths and the other Barbarian tribes. But the Roman Bishopric kept going, despite all that. Later it recovered, with Italy leading the Renaissance surge. It's been through thick and thin since then, sometimes attacked from without and sometimes attacked from within. But as Christ said, "The proud gates of Hell will not prevail against it".
As to the Papacy being regarded as the chief office, several of the earliest Church Fathers referred to it as just that -
Authority of the Pope — Church Fathers
Christ was setting up an office, and giving it authority when He spoke to Peter. It was my old Prostestant pastor who crystallised this in a conversation with me when he said, "Christ was setting up an office and giving it authority.... What's the use of having a church if you don't give it any authority!?"