Before we get into it, I wanna clarify that I’m not arguing against the historicity of the baptism or crufixion. I’m just arguing against a specific line of reasoning about it.
The criterion of embarrassment is the idea that a group would not make up false embarrassing facts about their own leaders, and therefore ideas such as “having to” get your sins baptized away or “succumbing to” a painful humiliating death could not have been made up or mythical.
But literary mythic characters like Daredevil and Luke Skywalker go through embarrassing and painful ordeals all the time. Humans do, and that’s reflected in human stories. And the Son of Man of the biblical story incarnated as just that, a human.
Especially since a lot of tellings, retellings, and exegeses of the story are meant to match up with the prophecy in Isaiah 53:12, which has him heading for that situation right from the get-go.