Here are two quotations explaining how the criteria of multiple attestation supposedly gives us a sound reason for believing in the historicity of a gospel account, the first by conservative Craig Evans and the second by liberal Bart Ehrman:
What about those who would like to have sound, compelling reasons for accepting the Gospel narratives as reliable? . . . Thoughtful people rightly apply criteria for evaluating claims. So also historians for assessing the historical worth of documents. . . . Sayings and actions of Jesus that appear in two or more independent sources suggest that they were circulated widely and early and were not invented by a single writer. . . . [This criteria enables] historians to give good reasons for judging this saying or that deed attributed to Jesus as authentic. (Fabricating Jesus, pp.49-51)
But what if a story is found independently in more than one source? That story cannot have been made up by either source, since they are independent; it must predate them both. Stories found in multiple, independent sources therefore have a better likelihood of being older, and possibly authentic. . . . For example, both Matthew and Luke independently indicate that Jesus was raised in Nazareth, but their stories about how he got there differ, so one came from M and the other from L. Mark indicates the same thing. So does John, which did not use any of the Synoptics or their sources. Conclusion? It is independently attested: Jesus probably came from Nazareth. (Jesus, Interrupted, p, 155)
And here’s a third from a quasi-legal religious text:
by the mouth of two or three witnesses the matter shall be established (Deuteronomy 19:5)
I like the third one, but the first two illustrate the logical fallacy of the false dichotomy or false dilemma. Granted the authors qualify their remarks with “suggest” and “probably”, but both consider only one set of alternative explanations for multiple attestation — unlikely coincidental fabrication or more likely genuine historicity.
Neither considers the possibility that independent sources could just as likely be independently addressing another theological debate or widely known unhistorical narrative.
Without attestation external to our gospel sources we have no way of knowing whether they were addressing historical events or other stories.
The only reason I can see for assuming the former and apparently giving no time for any other possibility is the desire to comply with popular religious and cultural belief systems.
The thousands of independent sitings of UFO’s do not establish that we really are being visited by aliens.