"Jesus" is an English invention, from Latin Iesus, from Greek Iesous, from Aramaic Yeshua (Y'shua), from Hebrew Yehoshua. Josephus just says "Iesous" the same as the Gospels do. "Yesou-" is the standard transliteration into Greek since they have no shin nor ayin, and "-s" indicates that it's a man's name.

On occasion writers might want to copy a word from a different language than they are writing in, in which case he could have tried to say "Yeshua, that is, Iesous"; but that could only be done either by (1) using a form of Hebrew alphabet, which Josephus knew but declined due to his audience, or (2) attempting a closer transliteration, which would still come out like "Iesoua" and would be relatively unnecessary. However, there's new strong statistical evidence (see the next comment I post) that Josephus's paragraph used Luke 24 as its source, which was Greek, so he might not have even thought to transliterate, the same as when he referred to any other person of the same name like Joshua ben Nun.

I downvoted Crock (but then reversed it), because he was falling for a seed of doubt that was sown in the 19th century and using a rationalization to perpetuate that doubt. There is no reason for doubt based on our distance from the texts, because when we pick up the culture of the texts the alleged reasons of distance disappear. The lack of vowels didn't cause anyone to have trouble reading the words "Yehoshua" and "Yeshua", with slight variation possible similar to other words. He was Yehoshua in the Hebrew synagogue, Yeshua in the Aramaic street, Yeshu in the Galilean gutter, Iesous in the Greek market, and Iesus in the Latin court.

Your link shows the word "yeshu'ah", salvation, which is close and cognate but not a homonym. People knew this related form (feminine) was similar, but when necessary they could emphasize the different ending to indicate which was meant if there was confusion.

u/CrockOSuds