Why not ジャシュ? It's much closer phonetically.
A couple reasons, the biggest being inertia. ジョシュア was how I was taught to write my name back in college, and it's just always been what I've used. I'm used to it as a name, so much so that if someone says something like, say, 女子, I'm inclined to turn my head as if someone's said my name. I have considered going with ジャシュ, but I have enough difficulty getting my mail properly addressed as it is. With ジョシュア, it's almost guaranteed to be spelled right in romaji, provided they use Hepburn style. I feel ジャシュ would lead to more confusion, and people will start writing notes and things to me with "Jash". (People who have heard my name in English, but not seen it written or pronounced in Japanese often do this anyway).
I spent a few decades writing my name in kana as ジョシュア, but when I had to get some business cards printed up a few years ago with my name in kana, I decided to go with ジャシュア as an experiment. ジョシュア for Joshua and ジョン for John probably get chosen by native English speakers because in romaji they become "joshua" and "jon", even though they are pronounced "joe-shoe-ah" and "joan".
That's pretty much how it is with me, although it wasn't a case of me choosing it as much as following established convention, and now, just being used to it.
The real issue is my last name. It leads to conversations like this:
Me: I'm Joshu Raiyaa.
Japanese person: Raiyaa? Like "usotsuki" (liar)?
Me: No! The pronunciation is completely different! My name means "sagi" (heron).
Japanese person: Oh! (lengthy pause) Sagi like "fraud"?
Me: The BIRD!!
I suppose to be really historically accurate, we should write our name the way the Joshua is written in the Old Testament in Japanese, which I think is ヨシュア.
Yup, that's how it's written. I like throwing folks for a loop by saying, "It's the same name that Jesus Christ had!" Their subsequent confusion allows me to spam them with a short course in cross-language phonological drift.