Who else would you tout as an early Christian writer that can be taken outside of the context of spreading early Xtian dogma?
Why would anyone want to write specifically about Christ in the first place apart from spreading Christian belief or denouncing it?
Tacitus mentions Jesus (real name Yeshua, but that's another day's discussion!) in passing and Latinises (is that a real word?) Christ to Christus. His mention:
"Christus, the founder of the name, was put to death by Pontius Pilate, procurator of Judea in the reign of Tiberius"
fairly well fits in with the description of Jesus, whilst his mention that Nero "falsely charged with the guilt [of burning Rome], and punished Christians, who were hated for their enormities", leaves us in no doubt about who is being referred to here, so we cannot leave aside Tacitus.
"The Elect" actually were the Christians - Christians being a derogatory term, and "the elect" being what they called themselves.
Regarding snake-worship, as Tacitus says, Christians were accused of all manner of bizarre and gross forms of worship by the Romans, from worshipping an
ass (as were the Jews) to cannibalism.
Josephus' Antiquities mentions Jesus twice: a short reference and a longer one.
One reference (Antiquities 20.9.1) is in his account of the execution of Jesus' brother James, where he is identified as "the brother of Jesus the so-called Christ, whose name was James".
The other (Antiquities 18.3.3) is as follows:
"Now there was about this time Jesus, a wise man, if it be lawful to call him a man, for he was a doer of wonderful works, a teacher of such men as receive the truth with pleasure. He drew over to him both many of the Jews, and many of the Gentiles. He was the Christ, and when Pilate, at the suggestion of the principal men among us, had condemned him to the cross, those that loved him at the first did not forsake him; for he appeared to them alive again the third day; as the divine prophets had foretold these and ten thousand other wonderful things concerning him. And the tribe of Christians so named from him are not extinct at this day."
Obviously this passage is not entirely genuine as here Josephus categorically calls Jesus "the Christ", whereas in the above cited passage he is "the so-called Christ".
However, scholars hold that some of the passage is genuine as it is in the style of Jospehus. "Now about this time..." being a favourite opener of his. What of the rest of it is genuine? Well if we remove all the references to Jesus as the Christ or as Divine we are left with a neutral description (as we have in elsewhere in Josephus of John the Baptist). We could go through it word by word with reference to the original Greek if you wish...
Some say that the passage must be entirely interpolated as it's not in the context of the surrounding passages. Readers of Josephus, however, and scholars, know Josephus to have been a patchy writer at best. One scholar described him as a writer of "inveterate sloppiness." His work is littered with seemingly out-of-context passages.
So what does Josephus tell us about Jesus?
Well firstly that he had a brother called James, who was an important Christian leader; that Jesus was virtuous and wise; that he had a following amongst Gentiles and Jews; that some saw him as the "Christ"; that he did surprising deeds (he doesn't specify that they were miracles); that he was crucified under Pilate, partly at the prompting of Jewish leaders; that his followers were called "Christians" after him (confirming Tacitus' terminology).
Other non-Christian writers making reference to Jesus or events surrounding his life are not as reliable as Tacitus and Josephus. These include Thallus, Pliny, Lucian, Suetonius, the Mara Bar-Serapion letter. And of course let's not forget the Talmud, the detraction of Jesus in which yields the name for thrash metallers "Pantera"!