I didn't know that he did. I think that is it agreed that He spoke in Aramaic, but we have no Aramaic Gospels. Odd, that.
Since the writers of the Gospels, at least 10s of years and perhaps 100s of years afterwards, wrote in Greek, then it makes sense to include the quotation in Greek, and the Greek quotation to use would be the LXX.
To do otherwise would have a mishmash of Greek, Aramaic, and Hebrew. Which would have been close to the historical reality, but far from what the literature that we have. That was my main argument with Mel Gibson's movie: They introduced Pontius Pilate's Latin, to what was, almost certainly a situation where Greek would have been spoken.
My other nit was that all the conversations were very very slow. I don't speak Aramaic, but my very limited smattering of Arabic would have played much faster. So would my Latin, if I had not done it in Greek.
It is my belief that the Qu'ran was written hundreds of years after the putative Mohammed, for contemporary political reasons (to justify the conquest), using the letters developed for Syrio Aramaic, and quite a few Syrio Aramaic words, that had no counterpart in Arabic. At the same time Sufi Muslims (who wore poor quality wool garments as a mark of their piety) were protesting against the "Court Islam" of their day. They were inspired by the austerity of Syriac Christian Monks.
Minor Irony: The "traditional woman's head cover"
made mandatory by the religious nutballs in Iran was introduced in Lebanon, and patented in 1979 by a gentleman who hoped to mark Muslim women as not being suitable targets for rape by Muslim/Palistinian thugs, inspired by the headgear of Maronite Christian Nuns.
Some Biblical scholars think that Matthew was originally written in Aramaic and later translated. Mark as Peter's secretary almost certainly spoke (and thought) in Aramaic, but his Gospel is believed to have been originally written in Greek.
I don't believe the general scholarship places the Gospels as late as you claim.
Of course the actors spoke fairly slowly - none of them were native speakers and they learned the languages for the movie! Next you'll be wanting the actors in spaghetti westerns to speak perfect English! The Latin was spoken by Pilate to other Romans such as his wife - he spoke Aramaic to Jesus, but Jesus replied in classical Latin (not the gutter Latin that the soldiers spoke.)