Historicists -- people who say there's no doubt Jesus existed (without necessarily making any supernatural claims about him) -- say that the number and the early date of the written witnesses to Jesus' existence are extremely impressive, and they're right. They go on to say that this makes Jesus' existence as certain as that of Socrates or Alexander the Great or Julius Caesar -- and they're wrong.
We have no contemporary written accounts of Jesus' life. Historicists promptly point out that we have no contemporary written sources for the life of Alexander either, and they're right. They point out further that the oldest surviving written account of Alexander comes nearly 300 years after his death, while Paul began to write the letters which became part of the New Testament with twenty years or so Jesus' death, and they're right again, assuming Jesus existed. However, the oldest surviving written accounts of Alexander which we have can be traced back clearly to accounts written by specifically-identified contemporaries. New Testament scholars are still looking for Q.
Unlike either Alexander or Jesus, several contemporary written accounts of both Socrates and Caesar exist, including books Caesar himself wrote. But it's not the number or date or even authorship of the written accounts of Jesus which separate him most decisively from Socrates, Alexander and Caesar. It's the quality of those writings. The preponderance of the supernatural. For the other three we have sources which concentrate on things they said and non-natural things they did like waging battles. Every early source of Jesus' life concentrates on the supernatural: miracles and resurrections and such.
We know what Socrates, Alexander and Caesar looked like, because we have sculptures which portray three identifiable people: an ugly, pot-bellied, balding, bearded guy; a handsome young man with a thick mane of curly hair and big spooky eyes (we know what Alexander's Dad looked like too); and a bald guy with a long skinny neck and thin lips, neither particularly handsome nor ugly.
Further evidence of Alexander's existence is several centuries' worth of Hellenistic culture from northwestern India to Egypt; of Caesar's, the Roman Empire. In their two cases the amount of history which would have to be un-written and then re-written to account for their non-existence is vast. In the case of both Socrates and Jesus, their impacts upon the world came entirely from the words of a small number of people who greatly admired them. Including, in Socrates' case, at least 2 specifically-indentifiable writers who knew him personally. Plus a 3rd contemporary writer who made fun of him. 0 contemporary writers in Jesus' case. And the earliest writer about Jesus saw him -- in a vision. Whatever that means.
We have no idea what Jesus looked like. Or Achilles. As in Jesus' case, no written account of Achilles' life is not so full of the supernatural that to describe his life with no supernatural elements doesn't require a complete re-write.
I think it's quite possible that there was an historical Achilles, a mighty Greek warrior who fought at Troy around the time that there might have been an historical Moses (about whose physical appearance we have no clue, the stories about whom are filled with the supernatural). I'm far from able to prove Achilles' historical existence. And yet an entire great culture, Socrates' culture, Alexander's culture, assumed that Achilles existed, and depended to a degree upon assumption like that. Alexander took a copy of the Iliad with him as he went conquering nations. He is supposed to have read and re-read it more than any other book. Even Caesar's culture, to a lesser degree, thought of Achilles as having been very real, and gave him a big role in their version of the history of the world.
I think that theories of an historical Achilles, of an historical Moses and an historical Jesus have much in common. Each of their stories is very important in the religious life of various civilizations. Each one of them might really have existed. Their stories had to begin somehow. Throw the story of King Arthur in there too, all of the above applies equally to him.
But leave Socrates and Alexander and Caesar out of this, because we know they existed.