Not quite. For starters, there are plenty of written records we can use to corroborate accounts of 1961. For seconds, the average life expectancy of people around that time is between 20 and 30 years (SOURCES:
https://books.google.co.uk/books?id=vh3pmAodawEC&pg=PA144&redir_esc=y#v=onepage&q&f=false https://books.google.co.uk/books?id=lvfMrkqDbY0C&pg=PA2&redir_esc=y#v=onepage&q&f=false).
Why would documents be produced saying that events
didn't occur?
Because people at the time wrote all kind of religious documents. It's not enough to say "they were risking death, so their accounts must be true!".
Not enough to establish that specific instances of his life should be accepted unequivocally as historical fact.
If you wish to make very specific claims about Homer's life, do you not think it reasonable to require contemporaneous sources?
Well, we retain quite a large number of works attributed to Caesar himself (
http://www.forumromanum.org/literature/caesarx.html), as well as lots of contemporary historical accounts from Cicero, Sallust, Nepos, Catallus, Pollio and Virgil.
A document being influential does not indicate the truth of its specific claims. Hence why so many theological texts exist that have millions, even billions, of adherents, other than the Bible.
Because people make very specific claims about their life, down to specific quotations, actions and beliefs. If you cannot corroborate any of this, you cannot assert it as historical fact. At most, you can conclude the existence of a figure called Jesus who played a pivotal role in the founding of the religious movement that came to be known as Christianity. Beyond that, it's basically heresay and storytelling.