to Neil,
No, they can't. T and J are late secondary sources that cannot corroborate anything from the early first century. The best they can do is inform us what T and J believed in their day. But of course there are also problematic issues with the authenticity even of that secondary evidence so they are even less useful than secondary evidence usually is for such inquiries.
The fact is the corroboration from Tacitus and Josephus are from non-Christians, who certainly had no interest to attest to someone who did not exist.
Tacitus and Josephus described events and mentioned persons happening/living before their time and most of them are not contested, even if sometimes uncorroborated. Why should it different in that case?
But Tacitus & Josephus corroborated in a different ways with each other about the existence of Jesus/Christ. And I do not see why there are problems with the authenticity of this secondary evidence. Mythicists do see problems, but that's because their half-baked theories would crash if they accept that evidence.
All we have is a document that says, in your words, that its author purports to say that he does not want readers to have a "worldly understanding of Jesus". I cannot see how that is evidence for a historical Jesus.
That "worldly understanding of Jesus" was administered in the past to the Corinthians by other(s) than Paul. And yes, that can be evidence for the past existence of a worldly Jesus. Even Paul was aware of that worldly Jesus:
1) His name is Jesus (Ro 5:15 "the one man Jesus Christ", 2 Cor 8:9, etc).
2)
He was a Jew (said to be descendant of Abraham (Gal 3:16), Israelites (Ro 9:4-5), Jesse (Ro 15:12) & David (Ro 1:3)).
3)
He was a minister/servant to (only) Jews (Ro 15:8).
4)
He was of no reputation and humble (Php 2:7-8).
5)
He was crucified (1 Cor 1:23, 2:2, 2:8, 2 Cor 13:4).
6)
The crucifixion happened in the heartland of the Jews: see here.
7)
He had brothers (contemporaries of Paul) (1 Cor 9:5).
8) These brothers were travelling with "a "sister", a wife" (1 Cor 9:5).
9)
One of Jesus' brothers was named "James" (Gal 1:19),
whom Paul met several times (Gal 1:19, 2:9).
10)
James lived for a long time in Jerusalem (Gal 1:19, 2:9).
11)
James was also an important member of some Jewish sect (Gal 2:2, 9, 12).
Notes:
a) I have been avoiding "crucified as Christ" because it is not clear, according to Paul, if HJ was considered "Christ" by any of his contemporaries before the crucifixion.
b) Also
Paul wrote Jesus was poor, in poverty (2 Cor 8:9) and
becoming of a woman (Gal 4:4).
c) Paul said Jesus was marked out (to show that someone or something is different from others (humans)) to be the Son of God ... by the resurrection from the dead" (Ro 1:4)
meaning Jesus did not provide any indication he was the Son of God prior to his alleged resurrection.
We cannot be confident of anything a single, uncorroborated source says. True. Yes, that means we cannot simply swallow much of Josephus naively. That means that we are limited in the sorts of things we can "know for a fact" in ancient history, true.
I think we can be fairly confident of uncorroborated sources, but only after they pass the test of critical analysis. If not, we would have to throw away a lot of accounts of the ancient historians.
You are describing the fallacious "nugget" theory of historical method. It's lots of fun. Really stimulates the imagination. Opens up many avenues for intellectual creativity. And as we see in your comment it even gives a sense of superiority to its practitioners. But it's simply all built on fallacious methods. It only produces muck that has been reshaped into something more pleasing to its author.
My example was about cold cases investigators. I don't think they are having fun by stimulating intellectual creativity (but in many cases, they produce good results).
I did not experience that when I developed my reconstruction. And when I let my creativity take hold of me, I soon discovered that was leading me astray, when the evidence, most of the time from more than one source, lead me to another direction.
And I was not after nuggets, selected by opinion or according to an agenda.
Cordially, Bernard