I really don't think the two sets of religious origins are identical, and i think the Frum example is frequently abused. Anyway the Gospel references are evidence. I have stated that to deny them to be evidence one would need to invoke a late origin forgery theory, and I note that you resort exactly to this.
The Gospels are NOT evidence for Jesus any more then Montanns' 1567
Discovery and plain Declaration of the Sundry Subtill practices of the Holy Inquisition Of Spain is evidence of what was going on in the Spanish Inquisition;
The Protocols of the Elders of Zion or The Protocols of the Meetings of the Learned Elders of Zion is evidence of a Jewish conspiracy; the Tanaka Memorial is evidence of what the Japanese military had planned in 1927; or countless other examples.
The Gospels are
propaganda with all that involves but the are NOT evidence, they are NOT history, they are NOT eyewitness testimony, or what every other song and dance one can come up with.
ETA we have a notice from Eusebius regarding the writing, now lost, of Papias, datable to before 120 AD. This is wiki.
I have pointed out before Eusebius also claimed "It is also recorded that under Claudius,
Philo came to Rome to have conversations with Peter, then preaching to the people there ... It is plain enough that he not only knew but welcomed with whole-hearted approval the apostolic men of his day, who it seems were of Hebrew stock and therefore, in the Jewish manner, still retained most of their ancient customs." (Eusebius,
The History of the Church, p50,52)
In
Ecclesiastical History Eusebius states that the Therapeutae (who Philo wrote of in his
The Contemplative Life) were the first
Christian monks.
So your golden boy Eusebius not only has Philo writing about the first
Christian monks but also has Philo having a nice talk with Peter in Rome and being well knowledgeable about "the apostolic men of his day" and yet the Philo we know wrote not one word on Jesus. Forgive the pun but how the hell does
that work?!?
Never mind Eusebius wrote in the 4th century and as pointed out by historian English historian Edward Gibbon over 200 years ago Eusebius himself wrote "That it will be necessary sometimes to use falsehood as a remedy for the benefit of those who require such a mode of treatment."
About 100 years later the cultural historian Jacob Burckhardt described Eusebius as "the first thoroughly dishonest historian of antiquity".