i'll be back in the morning to finish this up, but for quick notes:
Tacitus - not fond of christianity or christians, most christians would rather not cite him for reasons i will explain tomorrow
Suetonius - not favoured as citable by christians, mostly for having said "chrestus," more likely a jewish aggitator than Jesus Christ, especially since it unlikely he would have misspelled "christus" while spelling "christians" properly.
Thallus - did not once mention jesus directly (most definitely not his birth, as you say), most of his works are lost forever, did mention Moses (as did most other histiorians), and did mention a "darkness" that may have been related to the execution of Jesus (same time period).
Phlegon (of tralles, obviously) - generally ignored by most christians, despite having mentioned jesus, also spent more of his time writing about the existence of
ghosts,
prophecies by heads, monstrous births (Siamese twins), hemaphrodites and giant skeletons.
Serapion - you'll have to specify further, as notable serapions outside of christianity were physicians, and serapions within christianity were bishops or siants, or both.
Josephus - Jesus is mentioned in two passages of the work
The Antiquities of the Jews by the Jewish historian
Josephus, written in the late first century AD. One passage, known as the
Testimonium Flavianum,
discusses the career of Jesus. The authenticity of the Testimonium
Flavianum has been disputed since the 17th century, and by the mid 18th
century the consensus view was that it was at a minimum embellishment
by early Christian
scribes, if not a forgery. The other passage simply mentions Jesus as the brother of
James, also known as James the Just. Most scholars consider this passage genuine,
[1] but its authenticity has been disputed by
Emil Schürer as well by several recent popular writers. Josephus' other major work,
The Jewish War, makes no mention of Jesus.
if the pope(s) doesn't want to cite these people, why must other chistians?