Indeed the criterion of embarrassment.The gospel anecdotes involving his nuclear family are striking for the animosity evident in them, and so the criterion of embarrassment does indeed come in here.
His teachings on the ancient family are much less problematic in and of themselves, inasmuch as they cohere with his ethic of non-discriminating love beyond exclusive boundaries (i.e. not loving just those who love you and associating brotherhood/sisterhood not with blood but with being reborn in spirit of God) and family clans/tribes are inherently exclusive social groups.
However in a culture like ancient Israel, where the Torah literally defined that a person who insulted or disobeyed their parents was deserving of death, and in Roman civilization where the paterfamilias exercises sovereignty over his household, Jesus's behaviour was risque in the extreme.
It wouldn't have been something that any writer of that time, in my judgment, could have reasonably believed would've been a "good look" - that your cult's founder had rancid relations with his nearest and dearest.
So, I do tend to think that we have here a historical nugget, as the majority of scholars believe too.
And indeed it connects to the Torah's death penalty for disrespecting parents. which as I understand it had by then not be enforced for centuries, but which carried the opprobrium you mention.
Joseph is not mentioned by Paul or the authors of Mark or John, and his use in Matthew and Luke is rich in "fulfillment of prophecy" storytelling, which doesn't much dilute the general implication that Mary's household was pictured as having no husband figure (isn't Pantera's grave said to have been found in Germany?). If that's historical, it raises long-discussed views of Jesus' motives.