I would imagine that such a document, if authenticated, would be accepted as evidence for strengthening Christianity's authenticity. One accusation against the Bible, since several skeptics disregard the overwhelming evidence of the Gospel accounts, is that there are no contemporaneous (1st-c.) writing by someone who met him to authenticate that Jesus actually existed and did the things the Gospels claim. So I would imagine that such a discovery would bolster the case for Christianity's authenticity, since it would be an extra-biblical account of Jesus and at least a little bit about his life.

You too quickly brush off "the non-eyewitness testimony of the Gospels." You and I have probably discussed this before, but the Gospel of John, in particular, is strong with eyewitness details and elements. Despite that the earliest extant piece of John's Gospel (P52) is from AD 125-140, there is no evidence from the 1st-century that the authorship of John (or any of the Gospels) was ever in doubt.

I would not consider evidence of your new hypothetic document to be more compelling than the Gospel accounts because I regard the Gospel accounts to be deserving of far more respect and authority than you attribute to them.

If new fats about Jesus' historical existence were discovered today, how should Christians treat those facts? For example, suppose a first-century document was dug up from an archeological site and its date, writing style, and even its author was authenticated through modern historical methods, including corroborating evidence found in contemporary non-biblical literature. Further, our hypothetical document shows that the author claimed to have known of a man named Jesus, who he met in Galilee and was said to have demanded that all his fellow Jews should completely follow The Law and the Prophets. Should this newly found document be accepted as evidence for strengthening Christianity’s authenticity? Would evidence of this nature be more compelling than the non-eyewitness testimony of the Gospels?