I do not take White as an authority. But I would very hesitant to call her "false."
I take the writings of the "sect of the Nazoreans" that Eusubius decried and condemned as a better picture of what actually went on. They were the remnant of the Jewish Believers in Messiah. I have only seen part of the research. If you want more I suggest you look up the Messianic Jewish Theological Institute and ask around there for the results of the Bar Kochba revolt. Why the gentile church decided to change the sabbath to sunday, to do away with dietary restrictions, etc.
It was all to avoid persecution.
This too is not the whole picture. You have to look at both the ECF and the Ebionites, and the Nazarenes etc. Unfortunately most of the material of the latter is preserved only in the texts that treat them as heretics, such as Epiphanius (up until recently hard to get in an English translation), Irenaeus, etc.
I agree that the Nazarenes were likely the descendants of those who fled to Pella, and were following the Jewish ways of the Jerusalem church. These were Jews who received the Messiah, and practiced their faith in that way. Multiple sources refer to a gospel that was present with them, and some refer to Matthew particularly providing a gospel to them in the Hebrew language (which some take to be Aramaic, but let's not get into that now).
We also see early on though some hints that the Acts council was the actual impetus, not persecution. In Acts 21 James did NOT expect of the gentiles the same law observance as he did for the Jewish believers, both in Jerusalem and the diaspora. This was not due to persecution, but the decision handed down.
These groups continued to exist in harmony for some time, and later things went down hill. For instance in Justin martyr's time he still refers to those who hold to the same thought as James, observing the law, etc. as Christians, but now has come to hold them as deluded and weak minded. Essentially Justin would not have consigned James to hell, but would consider him heretical. Though of course, according to the council he was not heretical at all. Now by Chrysostom's time relations were off the charts bad between those Christians who endeavored to have any connection to Judaism. You can see a progression from Acts, to Justin, to Chrysostom. And while some defend Chrysostom saying he was using an example of hyperbole in his rhetoric, by nearly any measure he was incredibly hostile to those wanting anything to do with Judaism.
BTW - the results of that persecution are still on the front page today. Jews were forcibly removed from the Roman province of Judea and spread across the empire. Arabs from various tribes and locations were forcibly recruited to repopulate the province which Rome renamed "Palistina" (after the Philistines,) and then called the people "Palestinians."
There is no doubt that Hadrian's drastic measures caused some to further distance themselves from Jewish elements such as circumcision, etc. However, the gentile and Jewish practices were different long before that, and the decision in Acts allowed for that. One did not have to be circumcised and observe the law of Moses to be a Christian.
On the other hand, a Jewish Christian did not have to live like a gentile either, and that was so odd a notion that no one even needed to address it in Acts 15. However, James does raise the issue in Acts 21, due to what some were saying about Paul.
What worked well in the early church, each working to spread the message of Christ in their own sphere, started to break down later when there were fewer Jewish Christians around, and they tended to be isolated in a few areas. And the destruction of Jerusalem changed the centers of power in the church to gentile lands. The lack of contact between the groups was not good.
By the way, not all that Epiphanius or Iranaeus said about the Nazarenes was negative in any case. And their gospel was of some interest to various church fathers.
