The argument - that since there are no actual writings from the person in question proves they probably did not exist - is the weakest of all. Socrates did not write anything down and yet we do not hesitate to teach his philosophy from his followers writings. Plato...call the office.
Correction: Lack of evidence is not evidence of absence. No skeptic worth his salt would advance such a fallacious argument, and assert that "Christ did not leave any writings, ergo Christ did not exist."
As for Socrates: Here, the principle of "ordinary claims do not require extraordinary evidence, while extraordinary claims do require extraordinary evidence" prevails.
If Plato had gone on to describe how Socrates performed miracles, ascended bodily into Heaven, etc., then suddenly a higher standard of proof would have to be imposed.
Regards,