Yeah, cause it's reasonable to assume that the illiterate would write things down.
What precisely makes you think that Jesus and his twelve apostles were illiterate? Why would Jesus go unremarked on despite cuasing a huge fuss in Jerusalem, traveling everywehre with followers at his heels, and being tried by the governor of the province? Not only does he fail to write a damn thing, his followers, the "mulitudes" fail to write anything either, or even impress any chronicler, traveler, poet, politician, solider, or merchant enough to mention them for a generation.
I guess this means that Zoroaster didn't exist too, huh? Or that the Roman pagan religions didn't exist because they didn't write anything down until much later, right?
As for the pagan religions, Dionysus is certainly not historical. The historicity of Zoroaster is likewise doubtful. Are you making any claims that Hercules, or Mithra were real?
That Jesus et alia didn't write anything down isn't exceptional. It's expected. He was an itinerant preacher in a backwater providence of the Roman Empire. Literacy wasn't the norm for his class.
He was learned enough to astound the religous leaders of his town when he was a boy, remember? As for his "class" we know precious little about that. Either he was merely a tradesman for eighteen years, or he learned the oral and scriptural tradition of his people. Or, he turned into a trout for eighteen years. We don't know what learning he was supposed to have had because the four cannonical Gospels make no mention of eighteen years of his life.
I have no doubt Mohammed was an historical figure, despite doubting some claims about him. He is well documented by sources other than his followers. I have little doubt Apollonius of Tyana existed, though he almost certainly didn't travel to India, and he clearly couldn't walk through walls. The evidence for Jesus's existence is utterly absent. The earliest refernces to Jesus either fail to discuss his life at all, or discuss him as having existed at least a generation earlier. There is no contemporary mention of him anywhere by anyone.