Carroll's history is suspect. If he accepts those groups for one doctrine, then he gets to accept them for all their doctrines.
One thing about Carroll, though, that I think is worthy of note.
His intent was to show that one could trace the faith backward through history no matter what name those believers went by.
There is some merit to that notion.
It is the claim that Christianity can be identified by Christian teaching. The debate then becomes a discussion of the authoritative source of Christian teaching.
We say it is the words of the Apostles, founders of the Church, who were repeating the instruction they'd received from our Lord Jesus Himself.
I have trouble believing that anyone can disagree with that.
That is wisely stated, but Latter-Day-Saints and Jehova's Witnesses and Arians, and Bogomils and Gnostcis call on Christ and Apostoles' words, Gospels and "gospels."
His intent was to show that one could trace the faith backward through history no matter what name those believers went by
But it's not the same faith. Heresies existed alongside the Church.
His intent was to show that one could trace the faith backward through history no matter what name those believers went by.
FWIW, the names of these various Christian sects were mostly picked by those who were in opposition to them. So the same sect could end up with several different names through the course of history. Also, the derision that they are treated with is largely due to them not submitting to the state religion of the time. Also, what you find is one of the common denominators of these groups is a belief in the primacy of Scripture.