The “criteria of authenticity” were developed by New Testament scholars to establish a secure bedrock of information about the historical Jesus distinct from later Christian beliefs about him. The idea was that if a certain saying or deed attributed to Jesus could not be attributed to another Jewish or Christian source (double dissimilarity), was attested in different sources or traditional forms (multiple attestation), was translated from Aramaic, or ran counter to the theological agenda of the Christ followers who preserved the tradition (embarrassment), it likely derived from Jesus.
The validity of the criteria has been challenged. Lifting a bunch of sayings out of the literary contexts in which they were found, running each through a battery of tests to prove Jesus said it in isolation from the rest of the data, and recombining isolated sayings in new combinations to produce a novel reconstruction seems questionable as a historical method. I agree that the first step should be to ask how a certain reconstruction of Jesus could produce our diverse literary portraits of him in the Gospel texts as they stand. I am not ready to jettison the criteria entirely: I still think they are a useful tool for discerning older traditions incorporated into the Gospels (e.g. they show signs of translation or editorial activity or are attested in independent sources) and examining how they developed over time. I am aware, however, that the criteria are not purely objective measures that are not impacted by the subjective judgments of the interpreter who wields them.
An example can be found in the work of John Dominic Crossan. There is much I like in his The Historical Jesus – The Life of a Mediterranean Jewish Peasant: his thick description of the ancient social context, his attempt to be fair to non-canonical sources, his focus on the anti-Roman imperial dimensions of Jesus’ programme, and his transparent methodology. Crossan has a clear method in relying only on traditions he dates early, though this depends on some of Crossan’s problematic source-critical views and dating of texts, and that are multiply attested. Yet when the “Lord’s Prayer” passes the tests, Crossan writes “Still, despite the fact that the Lord’s prayer must be a very early summary of themes and emphases from Jesus own lifetime, I do not think that such a coordinated prayer was ever taught by him to his followers” (294). Could it be that the petition “your kingdom come” in the prayer suggests that Jesus anticipated the eschatological advent of the kingdom and runs counter to Crossan’s view that a Cynic-like sage was transformed into an apocalyptic prophet? Alternatively, the singly attested parable of the Good Samaritan is accepted into Crossan’s database.