UriYosef
So, unless you can provide supporting information to your #1, it must be regarded as a FALSE assumption, which puts into question your conclusions.

I must say you suprised me with this. I have to do some research. For now I say you may be right. And assuming you are right then my 'historical truth' 1 can be dropped. In that case my reasoning becomes:

1 Messiah's were coming and going; it was not a very rare phenomenon.
2 Crucifixion was a common punishment; many Jews and others were crucified for lots of reasons.
3 A religion (christianity) came into existence in the first century, centred (from hearsay) around a man called Yoshua/Jesus of Nazareth.

Conclusion:
It is very likely there was a man around 30 C.E. ...........
-who came from Nazareth,
-and who was called messiah by other people,
-and who was crucified by the Romans.

But then:
On the basis only of truth number 3 one may consider it very likely that there had been a man in the land of Israel that was crucified by the Romans, and who was percieved by a number of people to be the messiah.

Just a thought:
Could it be that the name Yeshua/Yoshua/Jesus as the supposed crucified messiah cannot be found in the Jewish sources you mentioned, because his name and story was not discussed by writers/Rabbi's, because of him supposedly being an idol. What I mean to say is: it was not discussed by the early Rabbi's, because of collective denial and not mentioning the name of an idol, so it was not written down in the sources (commentaries).
What we are doing here and now (Jew and christian discussing Torah and Jesus), is, to my knowledge, something unique in history since the first or 2nd century, and a result of developments after the 2nd WW. (I know there have been disputes, forced on Jews, with Clerical men, but that was exeptional, correct me if I'm wrong).
As I said just a thought.

Aad