The Church of Christ was invented by Alexander Campbell in the early Ninteenth Century. Theologically it is strongly influenced by the teachings of John Calvin.
I am not aware of any claim by the CoC (or any other protestant denomination) that their particular denomination has unbroken roots to the Apostolic age. Rather, the CoC (like many protestant churches) claim to practice religion in the same manner as described in the Bible (although no two denominations agree on exactly what that manner is).
It’s fine for the CoC to claim to be based upon the Bible, because the Bible existed when the denomination was first invented. But the Catholic Church could never claim to be based on the Bible, because the Bible had not yet been written when the Church first became active on Pentecost (well, none of the NT had been written, anyway).
So, if a denomination wants to be like the Apostolic Christians, they must not base themselves on the Bible. In fact, they must discard their Bibles, because Apostolic Christians did not have them either.
The various books that comprise the New Testament were not completed until about 70 years after Pentecost. But it’s not like the Church finally had a Bible in the year 110 or so. Other early Christian writings had been produced (such as the Shepherd of Hermes). There was no consensus on which writings should be viewed as Scriptural and which should not (Revelation was hotly contested, as you might imagine). There was quite a lot of bickering about the definition of the New Testament canon (we still have many writings of the Early Fathers arguing this topic).
There was no consensus on this until Pope Damascus accepted the proceedings of a Council of Carthage, which itself had accepted a Council of Hippo, which is the first time the list of NT books that we know today was ever enumerated. This did not happen until the early Sixth Century.
If you went to a Christian in the First, Second, Third, Fourth, or Fifth Century and asked to see a Bible, he would not know what you were talking about. The Bible (as we know it) did not exist (or at least the various NT books we know had never been put together and declared Scriptural). There was no recognized New Testament canon until the early Sixth Century.
The early Church had no Bible to base themselves on, and this was the case for centuries. So if any church wants to be like the early Christians, they must toss out their Bibles completely.