I suspect LH’s question was rhetorical, but I find it interesting. I have some thoughts about why it might be so widely loathed despite being almost universally practiced, none of which are meant as arguments that the usage is wrong.
First, I think literally is a bit different from some other Janus words in that “literally” seems to have an obvious connection to “literal”, and “literal”, AFAIK, is not used in a figurative fashion. This perhaps reinforces the notion that the use of literally that is consistent with “literal” is the right one, while the other use of literally is “wrong”, no matter how common it is. Again, I’m not suggesting that that is a good reason to reject the figurative usage, but I think it may help explain the pervasive negative reaction to it. With terms like “sanction” and “cleave”, OTOH, people don’t have any intuitve basis to seize on one of the usages as “correct” and the other as “incorrect,” so those terms don’t attract the same peevery.
Second, I think there is a concern that the figurative use of “literally” either has, or will, “ruin” the nonfigurative use of the term. I.e., the use of literally to flag a metaphorical statement has become so common that one’s first assumption in encountering it is that the statement is not literally true. I think this concern is somewhat grounded in reality: in an ambiguous sentence, using the word “literally” does little to clarify if a literal sense is meant or not, and, if anything, it would steer me towards the nonliteral one. But the concern is also a futile one: an attempt to close the barn door more than two hundred years after the horse escaped. “Language changes, deal with it,” is certainly a valid linguistic observation, but I can understand why it isn’t a satisfying answer, emotionally, to those who dislike this usage of the word. (Of course, it isn’t a linguistist’s job to give an emotionally satisfying explanation of the fact of language change to anybody.)
Finally, I find (and I suspect that this is true of others) that “literally”, used figuratively, annoys me when it is attached to an already annoying or poorly crafted bit of hyperbole, but I barely notice it when it is attached to a clever turn of phrase. For example, I cringed a bit when I heard a newscaster say there were “hordes of people, literally flinging their babies at the Pope.” I think it boils down to the fact that, for a variety of idiosyncratic reasons, the phrase “flinging babies at the Pope” annoyed me, and, when “literally” was attached to it, it made an annoying turn of phrase even worse. But when a skilled wordsmith uses literally in a figurative fashion, it doesn’t attract my peeving eye. And even pedestrian uses of “literally” often don’t bother me: being “glued to [one’s] seat” is a cliche, but, for whatever reason, it doesn’t annoy me, nor does the same phrase with “literally” attached to it. But, when an annoying metaphor or hyperbolic statement has “literally” attached to it, it is easy to fixate on the use of “literally,” and avoid the deeper analysis of what is, or isn’t, wrong with the turn of phrase itself. So, this may help explain why so many dispensers of writing advice condemn the figurative use of “literally”: it is easier to just tell students to avoid it than to give them nuanced guidance as to when it is perfectly fine and when it might best be avoided.