embarrassment (CoE), et al.
As tsig points out, what is "embarassing" morphs over time. The poster child for CoE is the baptism of Jesus by Dunker John.
Marcan Jesus, who is bucking only for the Messiah-ship, not for divinity (since he's Jewish, and the Jewish pantheon has a population of exactly one, with no children), gets a celebrity Jewish endorsement, plus the voice of God gives him a dustjacket blurb. There is no embarassment here; this is the cornerstone of the shaky case for the not-so-world-shaking Jesus being the one we've all been waiting for. As for sinfulness, John's batpism is (if Josephus has it right) a sign that you've already achieved rightness with God, not the means by which you achieve it. "Jesus is righteous." That's a boast, not a shameful admission.
As Jesus' ambitions rise in the later Gospels, and Christianity's own baptism becomes ritually efficacious (and a big part of what they sell), the baptism scene is retained and improved. There is a little more small talk, and so we have what "embarrassment" actually predicts, IMO, a song and dance to cover the points that might be "misunderstood." Note that it makes no difference whether the event actually happened or not: it is an established story, and as a story it has its uses (it explains why all Christians go swimming, which is not otherwise obvious).
And then there is John. Jesus is now God. So what happens to the baptism scene? It is dropped, without explanation or apology. What about the marketing finction of the story (you, too, should be baptized)? Jesus approves of what Dunker John is doing, even if he doesn't feel the need for a swim himself. And the big name Jewish endorsement? That's become Jesus and John endorsing each other - and ends with Jesus moving on to more important matters.
We could have a few laughs with the other criteria, too, but CoE is, I think, the epitome of absurdity. Now, if our boy Bart had said "Look, we need a filter for all these earth-toned Gospel stories. Patently self-serving BS should be treated with scepticism, and we'll only look twice at the rest" then we'd have something to discuss (beginning with if any of it is isn't self-serving, then why is it there at all?). But no, these "criteria," which are at best filters, Bart treats as touchstones - if a pericope passes the test(s) then we are supposedly certain that it is true - on pain of insulting professional historians and their Method.
Now, that's embarrassing, or it should be.