This is very interesting. Thank you for this discussion. I'm going to focus my response on three issues: (I) the promise of the criterion of embarrassment, (ii) the problem with the criterion of embarrassment, and (iii) the use of embarrassment to study other historical figures/events. (In other words, I'm going to leave the Mormon issue aside, though I appreciate how you're using it as a heuristic entrance into the question.)
First, the criterion of embarrassment is built upon a very useful and insightful observation (which I make forcefully in my STRUCTURING EARLY CHRISTIAN MEMORY; see pp. 59–61) that the past imposes itself in various ways onto the present. That is, we simply cannot ignore aspects of the past that we might wish to ignore, whether those aspects are traumatic, widely known, embarrassing, or whatever. The criterion of embarrassment rightly capitalizes on this point, noting that Jesus' tradents can hardly be imagined "fabricating a story where the Son of God was thought of getting his power from the devil" (as you say). If Jesus' opponents widely accused him of being aligned with the devil, then it is all the more likely that his followers would have to address that reputation and not be able to ignore it. This is the power of the criterion, and I hope it's clear that I am in no way refuting this point.
Second, however, the criterion of embarrassment is entirely impossible to apply. The fact is, we simply do not know what the Christians found "embarrassing" and what they did not. To stick with the example you offered (and which I quoted), the Beelzebul controversy does indeed accuse Jesus of demonic allegiance, but it also nicely tees up Jesus' opponents for a rhetorical butt-kicking that conclusively demonstrates the "arrival" of the kingdom of God (Matt 12.28||Luke 11:20) and Jesus' superiority over his opponents. If the charge of demonic possession and/or alliance was rhetorically useful for the evangelists, then how can we really say they demonstrate any particular "embarrassment" at the charge? The crucifixion provides an even better example: On the one hand, nothing is as embarrassing as a crucified messiah, but on the other hand nothing was as kerygmatically useful—even central—to Jesus' followers than his crucifixion (and even the biblical mandate for his crucifixion; e.g. Luke 24). This is the point of my essay: These data only become embarrassing when we plot them into a certain historical narrative (e.g. "Jesus' followers felt embarrassed by his baptism by and submission to John and so would never had created that tradition") rather than a different narrative (e.g. "Jesus' followers saw a chance to augment his reputation by linking him with the more popular prophet, John; only later, once Jesus' fame had eclipsed John, did his alleged baptism create problems"). But this is a critical problem: The criterion of embarrassment was supposed to identify facts that we could plot into an historical reconstruction, and now we see that the criterion itself already depends on a pre-established reconstruction in order to verify (or falsify) any particular datum.
Third, these problems are mitigated when we're studying other historical figures, not because they are qualitatively different from Jesus but because—often—we simply have much more historical data to work with. Let's imagine that someone wanted to apply something like the criterion of embarrassment to study the build-up to the invasion of Iraq in 2003. Obviously, the widespread agreement that Saddam Hussein had WMD, when contrasted with their post-invasion absence (I'm trying to keep my language neutral), could be perceived and discussed (and even verified) using the logic of embarrassment. But we have LOADS of evidence—from recordings and transcripts of Bush addressing the public, Colin Powell before the UN, reports from other nations (especially the UK), and even statements from Bush's predecessor, Bill Clinton—all of which confirming that in March 2003 America was adamant in its claim that Iraq had WMD and that this justified military action. With the gospels, by way of contrast, we don't have anything like this: no contemporary documents from Jesus' opponents expressing suspicion that Jesus was in league with Beelzebul, no reports from Jews scattered throughout the Roman Empire or Parthia claiming that a popular new prophet was widely considered demonic, not even parallel accounts of other Jewish figures having been popularly acclaimed exorcists but suspected of working with the prince of demons. So while these data might have been preserved in our gospels despite the embarrassment they caused Jesus' followers, we simply do not know. When studying other historical events/figures, especially in the modern era, we have more data available to verify this hypothesis.
Does that help?